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References

Published online by Cambridge University Press:  18 December 2014

Michael Baake  and
Uwe Grimm
Michael Baake
Affiliation:
Universität Bielefeld, Germany
Uwe Grimm
Affiliation:
The Open University, Milton Keynes
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Aperiodic Order , pp. 489 - 516
Publisher: Cambridge University Press
Print publication year: 2013

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References

[AW99] Adkins, W.A. and Weintraub, S.H. (1999). Algebra — An Approach via Module Theory, corr. 2nd printing (Springer, New York).Google Scholar
[Aki12] Akiyama, S. (2012). A note on aperiodic Ammann tiles, Discr. Comput. Geom. 48, 702–710.CrossRefGoogle Scholar
[AAP92] Alimov, Sh.A., Ashurov, R.R. and Pulatov, A.K. (1992). Multiple Fourier series and Fourier integrals. In Commutative Harmonic Analysis IV, Khavin, V.P. and Nikol'skiĭ, N.K. (eds.), pp. 1–95 (Springer, Berlin).Google Scholar
[A-PCG13] Aliste-Prieto, J., Coronel, D. and Gambaudo, J.-M. (2013). Linearly repetitive Delone sets are rectifiable, Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 275–290. arXiv:1103.5423.CrossRefGoogle Scholar
[All97] Allouche, J.-P. (1997). Schrödinger operators with Rudin-Shapiro potentials are not palindromic, J. Math. Phys. 38, 1843–1848.CrossRefGoogle Scholar
[ABCD03] Allouche, J.-P., Baake, M., Cassaigne, J. and Damanik, D. (2003). Palindrome complexity, Theor. Comput. Sci. 292, 9–31. arXiv:math.CO/0106121.CrossRefGoogle Scholar
[AMF95] Allouche, J.-P. and Mendès France, M. (1995). Automatic sequences. In [AG95], pp. 293–367.Google Scholar
[AS94] Allouche, J.-P. and Shallit, J. (1994). Complexité des suites de Rudin-Shapiro généralisés, Bull. Belg. Math. Soc. 1, 133–143.Google Scholar
[AS99] Allouche, J.-P. and Shallit, J. (1999). The ubiquitous Prouhet-Thue-Morse sequence. In Sequences and Their Applications: Proceedings of SETA '98, Ding, C., Helleseth, T. and Niederreiter, H. (eds.), pp. 1–16 (Springer, Berlin).Google Scholar
[AS03] Allouche, J.-P. and Shallit, J. (2003). Automatic Sequences (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[Alz95] Alzer, H. (1995). Note on an extremal property of the Rudin-Shapiro sequence, Abh. Math. Sem. Univ. Hamburg 65, 243–248.CrossRefGoogle Scholar
[AGS92] Ammann, R., Grünbaum, B. and Shephard, G.C. (1992). Aperiodic tiles, Discr. Comput. Geom. 8, 1–25.CrossRefGoogle Scholar
[AGZ10] Anderson, G.W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices (Cambridge University Press, Cambridge).Google Scholar
[AP98] Anderson, J.E. and Putnam, I.F. (1998). Topological invariants for substitution tilings and their associated C*-algebras, Ergodic Th. & Dynam. Syst. 18, 509–537.Google Scholar
[AAR99] Andrews, G.E., Askey, R. and Roy, R. (1999). Special Functions (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[Apo76] Apostol, T.M. (1976). Introduction to Analytic Number Theory (Springer, New York).Google Scholar
[Ata75] Ataman, Y. (1975). On positive definite measures, Monatsh. Math. 79, 265–272.CrossRefGoogle Scholar
[AP07] Au-Yang, H. and Perk, J.H.H. (2007). Q-dependent susceptibility in Z-invariant pentagrid Ising model, J. Stat. Phys. 127, 221–264. arXiv:cond-mat/0409557.Google Scholar
[AB09] Averkov, G. and Bianchi, G. (2009). Confirmation of Matheron's conjecture on the covariogram of planar convex bodies, J. Europ. Math. Soc. 11, 1187–1202. arXiv:0711.0572.Google Scholar
[AL12] Averkov, G. and Langfeld, B. (2012). On the reconstruction of planar lattice convex sets from the covariogram, Discr. Comput. Geom. 48, 216–238. arXiv:1011.5530.CrossRefGoogle Scholar
[AG95] Axel, F. and Gratias, D. (eds.) (1995). Beyond Quasicrystals (Springer, Berlin, and EDP Sciences, Les Ulis).CrossRef
[Baa84] Baake, M. (1984). Structure and representations of the hyperoctahedral group, J. Math. Phys. 25, 3171–3182.CrossRefGoogle Scholar
[Baa97] Baake, M. (1997). Solution of the coincidence problem in dimensions d ≤ 4. In [Moo97], pp. 9–44.Google Scholar
[Baa99] Baake, M. (1999). A note on palindromicity, Lett. Math. Phys. 49, 217–227. arXiv:math-ph/9907011.CrossRefGoogle Scholar
[Baa02a] Baake, M. (2002). A guide to mathematical quasicrystals. In [SSH02], pp. 17–48. arXiv:math-ph/9901014.Google Scholar
[Baa02b] Baake, M. (2002). Difiraction of weighted lattice subsets, Can. Math. Bull. 45, 483–498. arXiv:math.MG/0106111.CrossRefGoogle Scholar
[BB-A+94] Baake, M., Ben-Abraham, S.I., Klitzing, R., Kramer, P. and Schlottmann, M. (1994). Classification of local configurations in quasicrystals, Acta Cryst. A 50, 553–566.CrossRefGoogle Scholar
[BBM10] Baake, M., Birkner, M. and Moody, R.V. (2010). Difiraction of stochastic point sets: Explicitly computable examples, Commun. Math. Phys. 293, 611–660. arXiv:0803.1266.CrossRefGoogle Scholar
[BF05] Baake, M. and Frettlöh, D. (2005). SCD patterns have singular Difiraction, J. Math. Phys. 46, 033510: 1–10. arXiv:math-ph/0411052.CrossRefGoogle Scholar
[BFG07] Baake, M., Frettlöh, D. and Grimm, U. (2007). A radial analogue of Poisson's summation formula with applications to powder Difiraction and pinwheel patterns, J. Geom. Phys. 57, 1331–1343. arXiv:math.SP/0610408.CrossRefGoogle Scholar
[BG98] Baake, M. and Gähler, F. (1998). Symmetry structure of the Elser-Sloane quasicrystal. In Aperiodic '97, de Boissieu, M., Verger-Gaugry, J.-L. and Currat, R. (eds.), pp. 63–67 (World Scientific, Singapore). cond-mat/9809100.Google Scholar
[BGG12a] Baake, M., Gähler, F. and Grimm, U. (2012). Spectral and topological properties of a family of generalised Thue-Morse sequences, J. Math. Phys. 53, 032701: 1–24. arXiv:1201.1423.CrossRefGoogle Scholar
[BGG12b] Baake, M., Gähler, F. and Grimm, U. (2012). Hexagonal inflation tilings and planar monotiles, Symmetry 4, 581–602. arXiv:1210.3967.CrossRefGoogle Scholar
[BGG13] Baake, M., Gähler, F. and Grimm, U. (2013). Examples of substitution systems and their factors, J. Int. Seq. 16, article 13.2.14: 1-18); arXiv:1211.5466.Google Scholar
[BG97] Baake, M. and Grimm, U. (1997). Coordination sequences for root lattices and related graphs, Z. Krist. 212, 253–256. arXiv:cond-mat/9706122.Google Scholar
[BG03] Baake, M. and Grimm, U. (2003). A note on shelling, Discr. Comput. Geom. 30, 573–589. arXiv:math.MG/0203025.CrossRefGoogle Scholar
[BG07] Baake, M. and Grimm, U. (2007). Homometric model sets and window covariograms, Z. Krist. 222, 54–58. arXiv:math.MG/0610411.CrossRefGoogle Scholar
[BG08] Baake, M. and Grimm, U. (2008). The singular continuous Difiraction measure of the Thue-Morse chain, J. Phys. A: Math. Theor. 41, 422001: 1–6. arXiv:0809.0580.CrossRefGoogle Scholar
[BG09] Baake, M. and Grimm, U. (2009). Kinematic Difiraction is insufficient to distinguish order from disorder, Phys. Rev. B 79, 020203(R) and Phys. Rev. B 80, 029903(E). arXiv:0810.5750.CrossRefGoogle Scholar
[BG11] Baake, M. and Grimm, U. (2011). Kinematic Difiraction from a mathematical viewpoint, Z. Krist. 226, 711–725. arXiv:1105.0095.CrossRefGoogle Scholar
[BG12] Baake, M. and Grimm, U. (2012). Mathematical Difiraction of aperiodic structures, Chem. Soc. Rev. 41, 6821–6843. arXiv:1205.3633.CrossRefGoogle ScholarPubMed
[BG13] Baake, M. and Grimm, U. (2013). Squirals and beyond: Substitution tilings with singular continuous spectrum, Ergodic Th. & Dynam. Syst. 33, in press. arXiv:1205.1384.Google Scholar
[BGE97] Baake, M., Grimm, U. and Elser, V. (1997). The entropy of square-free words, Mathem. Comp. Modelling 26, 13–26. arXiv:math-ph/9809010.Google Scholar
[BGHZ08] Baake, M., Grimm, U., Heuer, M. and Zeiner, P. (2008). Coincidence rotations for the root lattice A4, Europ. J. Combin. 29, 1808–1819. arXiv:0709.1341.CrossRefGoogle Scholar
[BGJ93] Baake, M., Grimm, U. and Joseph, D. (1993). Trace maps, invariants, and some of their applications, Int. J. Mod. Phys. B 7, 1527–1550. arXiv:math- ph/9904025.CrossRefGoogle Scholar
[BGW94] Baake, M., Grimm, U. and Warrington, D. (1994). Some remarks on the visible points of a lattice, J. Phys. A: Math. Gen. 27, 2669–2674 and 5041 (erratum). arXiv:math-ph/9903046.CrossRefGoogle Scholar
[BHP97] Baake, M., Hermisson, J. and Pleasants, P.A.B. (1997). The torus parametrization of quasiperiodic LI classes, J. Phys. A: Math. Gen. 30, 3029–3056. mp_arc/02-168.CrossRefGoogle Scholar
[BHM08] Baake, M., Heuer, M. and Moody, R.V. (2008). Similar sublattices of the root lattice A4, J. Algebra 320, 1391–1408. arXiv:math.MG/0702448.CrossRefGoogle Scholar
[BH00] Baake, M. and Höffe, M. (2000). Difiraction of random tilings: Some rigorous results, J. Stat. Phys. 99, 219–261. arXiv:math-ph/9901008.CrossRefGoogle Scholar
[BJ90] Baake, M. and Joseph, D. (1990). Ideal and defective vertex configurations in the planar octagonal quasilattice, Phys. Rev. B 42, 8091–8102.CrossRefGoogle ScholarPubMed
[BJK91] Baake, M., Joseph, D. and Kramer, P. (1991). The Schur rotation as a simple approach to the transition between quasiperiodic and periodic phases, J. Phys. A: Math. Gen. 24, L961–967.CrossRefGoogle Scholar
[BJKS90] Baake, M., Joseph, D., Kramer, P. and Schlottmann, M. (1990). Root lattices and quasicrystals, J. Phys. A: Math. Gen. 23, L1037–L1041. arXiv:condmat/0006062.CrossRefGoogle Scholar
[BJS90] Baake, M., Joseph, D. and Schlottmann, M. (1990). The root lattice D4 and planar quasilattices with octagonal and dodecagonal symmetry, Int. J. Mod. Phys. B 5, 1927–1953.Google Scholar
[BKS92] Baake, M., Klitzing, R. and Schlottmann, M. (1992). Fractally shaped acceptance domains of quasiperiodic square-triangle tilings with dodecagonal symmetry, Physica A 191, 554–558.CrossRefGoogle Scholar
[BK11] Baake, M. and Kösters, H. (2011). Random point sets and their Difiraction, Philos. Mag. 91, 2671–2679. arXiv:1007.3084.CrossRefGoogle Scholar
[BKSZ90] Baake, M., Kramer, P., Schlottmann, M. and Zeidler, D. (1990). Planar patterns with fivefold symmetry as sections of periodic structures in 4-space, Int. J. Mod. Phys. B 4, 2217–2268.CrossRefGoogle Scholar
[BL04] Baake, M. and Lenz, D. (2004). Dynamical systems on translation bounded measures: Pure point dynamical and Difiraction spectra, Ergod. Th. & Dynam. Syst. 24, 1867–1893. arXiv:math.DS/0302061.CrossRefGoogle Scholar
[BL05] Baake, M. and Lenz, D. (2005). Deformation of Delone dynamical systems and pure point Difiraction, J. Fourier Anal. Appl. 11, 125–150. arXiv:math.DS/0404155.CrossRefGoogle Scholar
[BLM07] Baake, M., Lenz, D. and Moody, R.V. (2007). Characterisation of model sets by dynamical systems, Ergod. Th. & Dynam. Syst. 27, 341–382. arXiv:math.DS/0511648.CrossRefGoogle Scholar
[BLR07] Baake, M., Lenz, D. and Richard, C. (2007). Pure point Difiraction implies zero entropy for Delone sets with uniform cluster frequencies, Lett. Math. Phys. 82, 61–77. arXiv:0706.1677.CrossRefGoogle Scholar
[BM13] Baake, M. and Moll, M. (2013). Random noble means substitutions. In Aperiodic Crystals, Schmid, S., Withers, R.L. and Lifshitz, R. (eds.), pp. 19–27 (Springer, Dordrecht). arXiv:1210.3462.Google Scholar
[BM98] Baake, M. and Moody, R.V. (1998). Diffractive point sets with entropy, J. Phys. A: Math. Gen. 31, 9023–9039. arXiv:math-ph/9809002.CrossRefGoogle Scholar
[BM99] Baake, M. and Moody, R.V. (1999). Similarity submodules and root systems in four dimensions, Can. J. Math. 51, 1258–1276. arXiv:math.MG/9904028.CrossRefGoogle Scholar
[BM00a] Baake, M. and Moody, R.V. (2000). Self-similar measures for quasicrystals. In [BM00b], pp. 1–42. arXiv:math.MG/0008063.Google Scholar
[BM00b] Baake, M. and Moody, R.V. (eds.) (2000). Directions in Mathematical Quasicrystals, CRM Monograph Series, vol. 13 (AMS, Providence, RI).CrossRef
[BM04] Baake, M. and Moody, R.V. (2004). Weighted Dirac combs with pure point Difiraction, J. reine angew. Math. (Crelle) 573, 61–94. arXiv:math.MG/0203030.Google Scholar
[BMP00] Baake, M., Moody, R.V. and Pleasants, P.A.B. (2000). Difiraction for visible lattice points and kth power free integers, Discr. Math. 221, 3–42. arXiv:math.MG/9906132.CrossRefGoogle Scholar
[BMRS03] Baake, M., Moody, R.V., Richard, C. and Sing, B. (2003). Which distributions of matter diffract? — Some answers, In Quasicrystals: Structure and Physical Properties, H.-R., Trebin (ed.), Wiley-VCH, Berlin, pp. 188–207. arXiv:math- ph/0301019.Google Scholar
[BMS98] Baake, M., Moody, R.V. and Schlottmann, M. (1998). Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces, J. Phys. A: Math. Gen. 31, 5755–5765. arXiv:math-ph/9901008.CrossRefGoogle Scholar
[BSJ91] Baake, M., Schlottmann, M. and Jarvis, P.D. (1991). Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability, J. Phys. A: Math. Gen. 24, 4637–4654.CrossRefGoogle Scholar
[BS04a] Baake, M. and Sing, B. (2004). Kolakoski-(3, 1) is a (deformed) model set, Can. Math. Bull. 47, 168–190. arXiv:math.MG/0206098.CrossRefGoogle Scholar
[BS04b] Baake, M. and Sing, B. (2004). Difiraction spectrum of lattice gas models above Tc, Lett. Math. Phys. 68, 165–173. arXiv:math-ph/0405064.CrossRefGoogle Scholar
[BvE11] Baake, M. and van Enter, A.C.D. (2011). Close-packed dimers on the line: Difiraction versus dynamical spectrum, J. Stat. Phys. 143, 88–101. arXiv:1011.1628.CrossRefGoogle Scholar
[BW10] Baake, M. and Ward, T. (2010). Planar dynamical systems with pure Lebesgue Difiraction spectrum, J. Stat. Phys. 140, 90–102. arXiv:1003.1536.CrossRefGoogle Scholar
[BZ08] Baake, M. and Zint, N. (2008). Absence of singular continuous Difiraction for discrete multi-component particle models, J. Stat. Phys. 130, 727–740. arXiv:0709.2061.CrossRefGoogle Scholar
[Bak86] Bak, P. (1986). Icosahedral crystals from cuts in six-dimensional space, Scr. Metall. 20, 1199–1204.CrossRefGoogle Scholar
[Ban97] Bandt, C. (1997). Self-similar tilings and patterns described by mappings. In [Moo97], pp. 45–83.Google Scholar
[BW01] Bandt, C. and Wang, Y. (2001). Disk-like self-afine tiles in ℝ2, Discr. Comput. Geom. 26, 591–601.CrossRefGoogle Scholar
[BD07] Barge, M. and Diamond, B. (2007). Proximality in Pisot tiling spaces, Fund. Math. 194, 191–238. arXiv:math/0509051.CrossRefGoogle Scholar
[BDH03] Barge, M., Diamond, B. and Holton, C. (2003). Asymptotic orbits of primitive substitutions, Theor. Comput. Sci. 301, 439–450.CrossRefGoogle Scholar
[BO13] Barge, M. and Olimb, C. (2013). Asymptotic structure in substitution tiling spaces, Erg. Th. & Dynam. Syst. 33, in press. arXiv:1101.4902.Google Scholar
[BV11] Barnsley, M.F. and Vince, A. (2011). The chaos game on a general iterated function system, Erg. Th. & Dynam. Syst. 31, 1073–1079. arXiv:1005.0322.CrossRefGoogle Scholar
[Bau01] Bauer, H. (2001). Measure and Integration Theory, (de Gruyter, Berlin).CrossRefGoogle Scholar
[BF12] Bédaride, N. and Fernique, T. (2012). The Ammann-Beenker tilings revisited, Preprint arXiv:1208.3545.Google Scholar
[Bee82] Beenker, F.P.M. (1982). Algebraic theory of non-periodic tilings of the plane by two simple building blocks: A square and a rhombus, TH-Report 82-WSK-04 (TU Eindhoven).Google Scholar
[Bel95] Bellissard, J. (1995). Gap Labelling Theorems for Schrödinger's Operators. In From Number Theory to Physics, 2nd corr. printing, Waldschmidt, M., Moussa, P., Luck, J.-M. and Itzykson, C. (eds.), pp. 538–630 (Springer, Berlin).Google Scholar
[BHZ00] Bellissard, J., Herrmann, D.J.L. and Zarrouati, M. (2000). Hulls of aperiodic solids and gap labeling theorems. In [BM00b], pp. 207–258.Google Scholar
[BIST89] Bellissard, J., Iochum, B., Scoppola, E. and Testard, D. (1989). Spectral proper- ties of one-dimensional quasi-crystals, Commun. Math. Phys. 125, 527–543.CrossRefGoogle Scholar
[B-AG99] Ben-Abraham, S.I. and Gähler, F. (1999). Covering cluster description of octagonal MnSiAl quasicrystals, Phys. Rev. B 60, 860–864.CrossRefGoogle Scholar
[B-AQS13] Ben-Abraham, S.I., Quandt, A. and Shapira, D. (2013). Multidimensional paperfolding systems, Acta Cryst. A 69, 123–130.CrossRefGoogle ScholarPubMed
[BBD'E10] Benassi, C., Bianchi, G. and D'Ercole, G. (2010). Covariogram of non-convex sets, Mathematika 56, 267–284. arXiv:1003.4122.CrossRefGoogle Scholar
[Ben85] Bendersky, L. (1985). Quasicrystal with one-dimensional translational symmetry and a tenfold rotation axis, Phys. Rev. Lett. 55, 1461–1463.CrossRefGoogle Scholar
[BBK05] Benedetto, J.J., Bernstein, E. and Konstantinidis, I. (2005). Multiscale Riesz products and their support properties, Acta Appl. Math. 88, 201–227.CrossRefGoogle Scholar
[Ber65] Berberian, S.K. (1965). Measure and Integration (Macmillan, New York).Google Scholar
[BF75] Berg, C. and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups (Springer, Berlin).CrossRefGoogle Scholar
[Ber66] Berger, R. (1966). The undecidability of the domino problem, Memoirs Amer. Math. Soc. 66, 1–72.Google Scholar
[BD00] Bernuau, G. and Duneau, M. (2000). Fourier analysis of deformed model sets. In [BM00b], pp. 33–40.Google Scholar
[BFS12] Berthé, V., Frettlöh, D. and Sirvent, V. (2012). Selfdual substitutions in dimension one, Europ. J. Combin. 33, 981–1000. arXiv:1108.5053.CrossRefGoogle Scholar
[BS07] Berthé, V. and Siegel, A. (2007). Purely periodic beta-expansions in the Pisot non-unit case, J. Number Th. 153, 153–172. arXiv:math.DS/0407282.Google Scholar
[BSS+11] Berthé, V., Siegel, A., Steiner, W., Surer, P. and Thuswaldner, J.M. (2011). Fractal tiles associated with shift radix systems, Adv. Math. 226, 139–175. arXiv:0907.4872.CrossRefGoogle ScholarPubMed
[BDG+92] Bertin, M.J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M. and Schreiber, J.P. (1992). Pisot and Salem Numbers (Birkhäuser, Basel).CrossRefGoogle Scholar
[Bes54] Besicovitch, A.S. (1954) Almost Periodic Functions, reprint (Dover, New York).Google Scholar
[BKMS00] Bezuglyi, S., Kwiatkowski, J., Meydynets, K. and Solomyak, B. (2000). Invariant measures on stationary Bratteli diagrams, Ergod. Th. & Dynam. Syst. 30, 973–1007. arXiv:0812.1088.Google Scholar
[Bia02] Bianchi, G. (2002). Determining convex polygons from their covariograms, Adv. Appl. Prob. 34, 261–266.CrossRefGoogle Scholar
[Bia05] Bianchi, G. (2005). Matheron's conjecture for the covariogram problem, J. London Math. Soc. (2) 71, 203–220.CrossRefGoogle Scholar
[Bia09] Bianchi, G. (2009). The covariogram determines three-dimensional convex polytopes, Adv. Math. 220, 1771–1808.CrossRefGoogle Scholar
[BGK11] Bianchi, G., Gardner, R.J. and Kiderlen, M. (2011). Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms, J. Amer. Math. Soc. 24, 293–343. 1003.4486.CrossRefGoogle Scholar
[Bil95] Billingsley, P. (1995). Probability and Measure, 3rd ed. (Wiley, New York).Google Scholar
[Boh93] Bohl, P. (1893). Über die Darstellung von Functionen einer Variabeln durch trigonometrische Reihen mit mehreren einer Variabeln proportionalen Argumenten, Thesis (E.J. Karow, Dorpat).Google Scholar
[Boh47] Bohr, H. (1947). Almost Periodic Functions, reprint (Chelsea, New York).Google Scholar
[BT86] Bombieri, E. and Taylor, J.E. (1986). Which distributions of matter diffract? An initial investigation, J. Phys. Colloques 47, C3-19–C3-28.CrossRefGoogle Scholar
[BS66] Borevicz, Z.I. and Shafarevich, I.R. (1966). Number Theory (Academic Press, New York).Google Scholar
[Bou66] Bourbaki, N. (1966). General Topology, parts I and II (Hermann, Paris).Google Scholar
[Brl89] Brlek, S. (1989). Enumeration of factors in the Thue-Morse word, Discr. Appl. Math. 24, 83–96.CrossRefGoogle Scholar
[Bro65] Bromwich, T.J.I'A. (1965). An Introduction to the Theory of Infinite Series, 2nd rev. ed. (Macmillan, London).Google Scholar
[BBN+78] Brown, H., Bülow, R., Neubüser, J., Wondratschek, H. and Zassenhaus, H. (1978). Crystallographic Groups of Four-dimensional Space (Wiley, New York).Google Scholar
[BK98] Burago, D. and Kleiner, B. (1998). Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, Geom. Funct. Anal. 8, 273–282. arXiv:math.DG/9703022.CrossRefGoogle Scholar
[CJ94] Cabo, A.J. and Janssen, R.H.P. (1994). Cross-covariance functions characterise bounded regular closed sets, CWI Report BS-R9426, available at http://oai.cwi.nl/oai/asset/5110/5110D.pdfGoogle Scholar
[CIV03] Campanino, M., Ioffe, D. and Velenik, Y. (2003). Ornstein-Zernike theory for the finite range Ising models above Tc, Probab. Th. Relat. Fields 125, 305–349. arXiv:math.PR/0111274.CrossRefGoogle Scholar
[CL90] Carmona, R. and Lacroix, J. (1990). Spectral Theory of Random Schrödinger Operators (Birkhäuser, Boston, MA).CrossRefGoogle Scholar
[Car00] Carothers, N.L. (2000). Real Analysis (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[Cas71] Cassels, J.W.S. (1971). An Introduction to the Geometry of Numbers, 2nd corr. printing (Springer, Berlin).Google Scholar
[CS13] Cellarosi, F. and Sinai, Ya.G. (2013). Ergodic properties of square-free numbers, J. Europ. Math. Soc. 15, 1343–1374. arXiv:1112.4691.CrossRefGoogle Scholar
[Cha87] Champeney, D.C. (1987). A Handbook of Fourier Theorems (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[CMP98] Chen, L., Moody, R.V. and Patera, J. (1998). Non-crystallographic root systems. In [Pat98], pp. 135–178.Google Scholar
[CSM88] Cheng, Z., Savit, R. and Merlin, R. (1988), Structure and electronic properties of Thue-Morse lattices, Phys. Rev. B 37, 4375–4382.CrossRefGoogle ScholarPubMed
[CS06] Clark, A. and Sadun, L. (2006). When shape matters: Deformation of tiling spaces, Ergod. Th. & Dynam. Syst. 26, 69–86. arXiv:math.DS/0306214.CrossRefGoogle Scholar
[CE80] Collet, P. and Eckmann, J.-P. (1980). Iterated Maps on the Interval as Dynamical Systems (Birkhäuser, Basel).Google Scholar
[CT08] Connelly, B. and Terrell, B. (2008). Highly Symmetric Tensegrity Structures, available at http://www.math.cornell.edu/ tens/Google Scholar
[CKH98] Conrad, M., Krumeich, F. and Harbrecht, B. (1998). A dodecagonal quasicrystalline chalcogenide, Angew. Chem. Int. Ed. 37, 1383–1386.3.0.CO;2-R>CrossRefGoogle ScholarPubMed
[CBG08] Conway, J.H., Burgiel, H. and Goodman-Strauss, C. (2008). The Symmetries of Things (A.K. Peters, Wellesley, MA).Google Scholar
[CR98] Conway, J.H. and Radin, C. (1998). Quaquaversal tilings and rotations, Invent. Math. 132, 179–188.CrossRefGoogle Scholar
[CS99] Conway, J.H. and Sloane, N.J.A. (1999). Sphere Packings, Lattices and Groups, 3rd ed. (Springer, New York).CrossRefGoogle Scholar
[Cor89] Corduneanu, C. (1989). Almost Periodic Functions, 2nd English ed. (Chelsea, New York).Google Scholar
[CFS82] Cornfeld, I.P., Fomin, S.V. and Sinai, Ya.G. (1982). Ergodic Theory (Springer, New York).CrossRefGoogle Scholar
[CDP10] Cortez, M.I., Durand, F. and Petite, S. (2010). Linearly repetitive Delone systems have a finite number of non-periodic Delone system factors, Proc. Amer. Math. Soc. 138, 1033–1046. arXiv:0807.2907.CrossRefGoogle Scholar
[CS11] Cortez, M.I. and Solomyak, B. (2011). Invariant measures for non-primitive tiling substitutions, J. d'Analyse Math. 115, 293–342. arXiv:1007.1686.CrossRefGoogle Scholar
[CH73] Coven, E.M. and Hedlund, G.A. (1973). Sequences with minimal block growth, Math. Systems Theory 7, 138–153.CrossRefGoogle Scholar
[Cow95] Cowley, J.M. (1995). Difiraction Physics, 3rd ed. (North-Holland, Amsterdam).Google Scholar
[Cox73] Coxeter, H.S.M. (1973). Regular Polytopes, 3rd ed. (Dover, New York).Google Scholar
[CEPT86] Coxeter, H.S.M., Emmer, M., Penrose, R. and Teuber, M.L. (eds.) (1986). M.C. Escher. Art and Science (North-Holland, Amsterdam).
[CM80] Coxeter, H.S.M. and Moser, W.O.J. (1980). Generators and Relations for Discrete Groups, 4th ed. (Springer, Berlin).CrossRefGoogle Scholar
[Cul96] Culik, K. (1996). An aperiodic set of 13 Wang tiles, Discr. Math. 160, 245–251.CrossRefGoogle Scholar
[Cur84] Curtis, M.L. (1984). Matrix Groups, 2nd ed. (Springer, New York).CrossRefGoogle Scholar
[DV-J88] Daley, D.D. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes (Springer, New York).Google Scholar
[Dam01] Damanik, D. (2001). Schrödinger operators with low-complexity potentials, Ferroelectrics 250, 143–149.CrossRefGoogle Scholar
[DL99] Damanik, D. and Lenz, D. (1999). Uniform spectral properties of onedimensional quasicrystals, I. Absence of eigenvalues, Commun. Math. Phys. 207, 687–696. arXiv:math-ph/9903011.CrossRefGoogle Scholar
[DL06a] Damanik, D. and Lenz, D. (2006). Substitution dynamical systems: Characterization of linear repetitivity and applications, J. Math. Anal. Appl. 321, 766–780. arXiv:math.DS/0302231.CrossRefGoogle Scholar
[DL06b] Damanik, D. and Lenz, D. (2006). Zero-measure Cantor spectrum for Schrödinger operators with low-complexity potentials, J. Math. Pures Appl. 85, 671–686.CrossRefGoogle Scholar
[Dan89] Danzer, L. (1989). Three-dimensional analogs of the planar Penrose tilings and quasicrystals, Discr. Math. 76, 1–7.CrossRefGoogle Scholar
[Dan95] Danzer, L. (1995). A family of 3D-spacefillers not permitting any periodic or quasiperiodic tiling. In Aperiodic '94, Chapuis, G. and Paciorek, W. (eds.), pp. 11–17 (World Scientific, Singapore).Google Scholar
[DPT93] Danzer, L., Papadopolos, Z. and Talis, A. (1993). Full equivalence between Socolar's tilings and the (A,B,C,K)-tilings leading to a rather natural decoration, Int. J. Mod. Phys. B 7, 1379–1386.CrossRefGoogle Scholar
[DSvOD93] Danzer, L., Sonneborn, P., van Ophuysen, G. and Duitmann, S. (1993). The {A,B,C,K}-Book (Internal Report, Univ.Dortmund).Google Scholar
[dBr81] de Bruijn, N.G. (1981). Algebraic theory of Penrose's non-periodic tilings of the plane. I & II, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 84, 39–52 and 53–66.Google Scholar
[dGN97a] de Gier, J. and Nienhuis, B. (1997). The exact solution of an octagonal rectangle triangle random tiling, J. Stat. Phys. 87, 415–437. arXiv:solv-int/9610009.CrossRefGoogle Scholar
[dGN97b] de Gier, J. and Nienhuis, B. (1997). On the integrability of the square-triangle random tiling model, Phys. Rev. E 55, 3926–3933. arXiv:solv-int/9611005.CrossRefGoogle Scholar
[dGN98] de Gier, J. and Nienhuis, B. (1998). Bethe Ansatz solution of a decagonal rectangle-triangle random tiling, J. Phys. A: Math. Gen. 31, 2141–2154. arXiv:cond-mat/9709338.Google Scholar
[Dek78] Dekking, F.M. (1978). The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitsth. verw. Geb. 41, 221–239.CrossRefGoogle Scholar
[Dek97] Dekking, F.M. (1997). What is the long range order in the Kolakoski sequence? In [Moo97], pp. 115–125.Google Scholar
[DM09] Deng, X. and Moody, R.V. (2009). How model sets can be determined by their two-point and three-point correlations. J. Stat. Phys. 135, 621–637. arXiv:0901.4381.CrossRefGoogle Scholar
[DGS76] Denker, M., Grillenberger, C. and Sigmund, K. (1976). Ergodic Theory on Compact Spaces, LNM 527 (Springer, Berlin).CrossRefGoogle Scholar
[dSp99] de Spinadel, V.W. (1999). The family of metallic means, Visual Math. 1, issue 3, paper 4; available online as http://www.mi.sanu.ac.rs/vismath/spinadel/index.html.Google Scholar
[DSS95] Deuber, W.A., Simonovits, M. and Sós, V.T. (1995). A note on paradoxical metric spaces, Stud. Sci. Math. Hung. 30, 17–23; revised and annotated version available at http://mta.renyi.hu/~miki/walter07.pdf.Google Scholar
[Dev89] Devaney, R.L. (1989). An Introduction to Chaotic Dynamical Systems, 2nd ed. (Addison-Wesley, Redwood City).Google Scholar
[dWo74] de Wolff, P.M. (1974). The pseudo-symmetry of modulated crystal structures, Acta Cryst. A 30, 777–785.CrossRefGoogle Scholar
[Die70] Dieudonné, J. (1970). Treatise of Analysis, vol. II (Academic Press, New York).Google Scholar
[Dod83] Dodd, F.W. (1983). Number Theory in the Quadratic Field with Golden Section Unit (Polygonal Publishing House, Passaic, NJ).Google Scholar
[DG72] Domb, C. and Green, M.S. (eds.) (1972). Phase Transitions and Critical Phenomena. Vol. 1. Exact Results (Academic Press, London).
[DSV00] Draco, B., Sadun, L. and Van Wieren, D. (2000). Growth rates in the quaquaversal tiling, Discr. Comput. Geom. 23, 419–435. arXiv:math-ph/9812018.CrossRefGoogle Scholar
[Dur00] Durand, F. (2000). Linearly recurrent subshifts have a finite number of nonperiodic subshift factors, Ergod. Th. & Dynam. Syst. 20, 1061–1078 and 23 (2003), 663–669 (corrigendum and addendum). arXiv:0807.4430.CrossRefGoogle Scholar
[Dwo93] Dworkin, S. (1993). Spectral theory and X-ray Difiraction, J. Math. Phys. 34, 2965–2967.CrossRefGoogle Scholar
[DMcK72] Dym, H. and McKean, H.P. (1972). Fourier Series and Integrals (Academic Press, New York).Google Scholar
[Dys62] Dyson, F. (1962). Statistical theory of the energy levels of complex systems. III, J. Math. Phys. 3, 166–175.Google Scholar
[Ebe02] Ebeling, W. (2002). Lattices and Codes, 2nd ed. (Vieweg, Braunschweig).CrossRefGoogle Scholar
[ENP07] Eigen, S.Navarro, J. and Prasad, V. S. (2007). An aperiodic tiling using a dynamical system and Beatty sequences. In Dynamics, Ergodic Theory, and Geometry, Hasselblatt, B. (ed.), MSRI Publications, vol. 54, pp. 223–241 (Cambridge University Press, Cambridge).Google Scholar
[EW11] Einsiedler, M. and Ward, T. (2011). Ergodic Theory, with a View towards Number Theory (Springer, London).Google Scholar
[Els85] Elser, V. (1985). Comment on “Quasicrystals: A new class of ordered structures”, Phys. Rev. Lett. 54, 1730.CrossRefGoogle Scholar
[Els86] Elser, V. (1986). The Difiraction pattern of projected structures, Acta Cryst. A 42, 36–43.CrossRefGoogle Scholar
[ES87] Elser, V. and Sloane, N.J.A. (1987). A highly symmetric four-dimensional quasicrystal, J. Phys. A: Math. Gen. 20, 6161–6168.CrossRefGoogle Scholar
[Esc04] Esclangon, E. (1904). Les fonctions quasi-périodiques, Thesis (Gauthier-Villars, Paris).Google Scholar
[Ete81] Etemadi, N. (1981). An elementary proof of the strong law of large numbers, Z. Wahrscheinlichkeitsth. verw. Geb. 55, 119–122.CrossRefGoogle Scholar
[Fek23] Fekete, M. (1923). Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koefizienten, Math. Z. 17, 228–249.CrossRefGoogle Scholar
[Fel72] Feller, W. (1972). An Introduction to Probability Theory and Its Applications, vol. II, 2nd ed. (Wiley, New York).Google Scholar
[FeS12] Fernique, T. and Sablik, M. (2012). Local rules for computable planar tilings, EPTCS 90, 133–141. arXiv:1208.2759.CrossRefGoogle Scholar
[FR03] Fisher, B.N. and Rabson, D.A. (2003). Applications of group cohomology to the Classification of quasicrystal symmetries, J. Phys. A: Math. Gen. 36, 10195–10214. arXiv:math-ph/0105010.CrossRefGoogle Scholar
[For10] Forrester, P.J. (2010). Log-Gases and Random Matrices, (Princeton University Press, Princeton).Google Scholar
[Fra97] Fraczek, K.M. (1997). On a function that realizes the maximal spectral type, Studia Math. 124, 1–7.CrossRefGoogle Scholar
[Fra03] Frank, N.P. (2003). Substitution sequences in ℤd with a non-simple Lebesgue component in the spectrum, Ergod. Th. & Dynam. Syst. 23, 519–532.CrossRefGoogle Scholar
[Fra05] Frank, N.P. (2005). Multi-dimensional constant-length substitution sequences, Topol. Appl. 152, 44–69.CrossRefGoogle Scholar
[Fra08] Frank, N.P. (2008). A primer of substitution tilings of the Euclidean plane, Expo. Math. 26, 295–326. arXiv:/0705.1142.CrossRefGoogle Scholar
[Fra09] Frank, N.P. (2009). Spectral theory of bijective substitution sequences, MFO Reports 6, 752–756.Google Scholar
[FR08] Frank, N.P. and Robinson, E.A. (2008). Generalized β-expansions, substitution tilings, and local finiteness, Trans. Amer. Math. Soc. 360, 1163–1177. arXiv:math.DS/0506098.CrossRefGoogle Scholar
[FS09] Frank, N.P. and Sadun, L. (2009). Topology of some tiling spaces without finite local complexity, Discr. Cont. Dyn. Syst. A 23, 847–865. arXiv:math.DS/0701424.Google Scholar
[FS12] Frank, N.P. and Sadun, L. (2012). Fusion tilings with Infinite local complexity, Preprint arXiv:1201.3911.Google Scholar
[FSo01] Frank, N.P. and Solomyak, B. (2001). A characterization of planar pseudo-self-similar tilings, Discr. Comput. Geom. 26, 289–306.Google Scholar
[FW11] Frank, N.P. and Whittaker, M.F. (2011). A fractal version of the pinwheel tiling, Math. Intelligencer 33 (2), 7–17. arXiv:1001.2203.CrossRefGoogle Scholar
[Fre02] Frettlöh, D. (2002). Nichtperiodische P?asterungen mit ganzzahligem Inflationsfaktor, PhD thesis (Univ. Dortmund).Google Scholar
[Fre08] Frettlöh, D. (2008). Substitution tilings with statistical circular symmetry, Europ. J. Combin. 29, 1881–1893. arXiv:0704.2521.CrossRefGoogle Scholar
[Fre11] Frettlöh, D. (2011). A fractal fundamental domain with 12-fold symmetry, Symmetry: Culture and Science 22, 237–246.Google Scholar
[Fre13] Frettlöh, D. (2013). Highly symmetric fundamental cells for lattices in ℝ2 and ℝ3, Preprint arxiv:1305.1798.Google Scholar
[FG13] Frettlöh, D. and Garber, A. (2013). Bi-Lipschitz equivalence and wobbling equivalence of Delone sets, in preparation.Google Scholar
[FH13] Frettlöh, D. and Harriss, E. (2013). Parallelogram tilings, worms, and finite orientations, Discr. Comput. Geom. 49, 531–539. arXiv:1202.4686.CrossRefGoogle Scholar
[FR13] Frettlöh, D. and Richard, C. (2013). Dynamical properties of almost repetitive Delone sets, Discr. Cont. Dynam. Syst. A 33, in press. arXiv:1210.2955.Google Scholar
[FSi07] Frettlöh, D. and Sing, B. (2007). Computing modular coincidences for substitution tilings and point sets, Discr. Comput. Geom. 37, 381–407. arXiv:math.MG/0601067.CrossRefGoogle Scholar
[Gäh88] Gähler, F. (1988). Quasicrystal Structures from the Crystallograhic Viewpoint, PhD thesis no. 8414 (ETH Zürich).Google Scholar
[Gäh93] Gähler, F. (1993). Matching rules for quasicrystals: The composition-decomposition method, J. Non-Cryst. Solids 153&154, 160–164.Google Scholar
[Gäh10] Gähler, F. (2010). MLD relations of Pisot substitution tilings, J. Phys.: Conf. Ser. 226, 012020: 1–6.Google Scholar
[Gäh13] Gähler, F. (2013). Substitution rules and topological properties of the Robinson tilings. In Aperiodic Crystals, Schmid, S., Withers, R.L. and Lifshitz, R. (eds.), pp. 67–73 (Springer, Dordrecht). arXiv:1210.6468.Google Scholar
[GGB03] Gähler, F., Gummelt, P. and Ben-Abraham, S.I. (2003). Generation of quasiperiodic order by maximal cluster covering. In [KP03], pp. 63–95.Google Scholar
[GJS12] Gähler, F., Julien, A. and Savinien, J. (2012). Combinatorics and topology of the Robinson tiling, Preprint arXiv:1203.1387.Google Scholar
[GK97] Gähler, F. and Klitzing, R. (1997). The Difiraction pattern of self-similar tilings. In [Moo97], pp. 141–174.Google Scholar
[GM13] Gähler, F. and Maloney, G. (2013). Cohomology of one-dimensional mixed substitution tiling spaces, Topol. Appl. 160, 703–719. arXiv:1112.1475.CrossRefGoogle Scholar
[GR86] Gähler, F. and Rhyner, J. (1986). Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings, J. Phys. A: Math. Gen. 19, 267–277.CrossRefGoogle Scholar
[GS93] Gähler, F. and Stampfli, P. (1993). The dualisation method revisited: Dualisation of product Laguerre complexes as a unifying framework, Int. J. Mod. Phys. B 6–7, 1333–1349.Google Scholar
[Gan59] Gantmacher, F.R. (1959). The Theory of Matrices, reprint (AMS Chelsea, New York).Google Scholar
[Gar77] Gardner, M. (1977). Extraordinary nonperiodic tiling that enriches the theory of tiles, Sci. Amer. 236, 110–121.Google Scholar
[GGZ05] Gardner, R.J., Gronchi, P. and Zong, C. (2005). Sums, projections, and sections of lattice sets, and the discrete covariogram, Discr. Comput. Geom. 34, 391–409.CrossRefGoogle Scholar
[Geo11] Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd ed. (de Gruyter, Berlin).CrossRefGoogle Scholar
[Geo12] Georgii, H.-O. (2012). Stochastics: An Introduction to Probability and Statistics, 2nd ed. (de Gruyter, Berlin).CrossRefGoogle Scholar
[GLA90] Gil de Lamadrid, J. and Argabright, L.N. (1990). Almost periodic measures, Memoirs Amer. Math. Soc. 65, no. 428 (AMS, Providence, RI).Google Scholar
[Gin65] Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phys. 6, 440–449.CrossRefGoogle Scholar
[Gli10] Glied, S. (2010). Coincidence and Similarity Isometries of Modules in Euclidean Space, PhD thesis (Univ. Bielefeld).Google Scholar
[Gne98] Gnedenko, B.V. (1998). Theory of Probability, 6th ed. (CRC press, Amsterdam).Google Scholar
[God89] Godrèche, C. (1989). The sphinx: A limit-periodic tiling of the plane, J. Phys. A: Math. Gen. 22, L1163–L1166.CrossRefGoogle Scholar
[GL89] Godrèche, C. and Luck, J.M. (1989). Quasiperiodicity and randomness in tilings of the plane, J. Stat. Phys. 55, 1–28.CrossRefGoogle Scholar
[Gol49] Golay, M.J.E. (1949). Multislit spectroscopy, J. Opt. Soc. Amer. 39, 437–444.Google Scholar
[Goo98] Goodman-Strauss, C. (1998). Matching rules and substitution tilings, Ann. Math. 147, 181–223.CrossRefGoogle Scholar
[Goo99a] Goodman-Strauss, C. (1999). A small aperiodic set of planar tiles, Europ. J. Combin. 20, 375–384.CrossRefGoogle Scholar
[Goo99b] Goodman-Strauss, C. (1999). An aperiodic pair of tiles in En for alls n ≥ 3, Europ. J. Combin. 20, 385–395.CrossRefGoogle Scholar
[Goo03] Goodman-Strauss, C. (2003). Matching rules for the sphinx substitution tiling, unpublished notes, available at http://comp.uark.edu/strauss/Google Scholar
[Goo11] Goodman-Strauss, C. (2011). Tazzellazioni. In La Matematica vol III: Suoni, Forme, Parole, Bartocci, C. and Odifreddi, P. (eds.), pp. 249–285 (Einaudi, Torino). English version: Tessellations, available at http://mathfactor.uark.edu/downloads/tessellations.pdfGoogle Scholar
[Gou03] Gouéré, J.-B. (2003). Difiraction and Palm measure of point processes, C. R. Acad. Sci. Paris 342, 141–146. arXiv:math.PR/0208064.Google Scholar
[Gou05] Gouéré, J.-B. (2005). Quasicrystals and almost periodicity, Commun. Math. Phys. 255, 655–681. arXiv:math-ph/0212012.CrossRefGoogle Scholar
[Gou97] Gouvêa, F.Q. (1997). p-adic Numbers — An Introduction, 2nd ed. (Springer, Berlin).CrossRefGoogle Scholar
[GKQ95] Gratias, D., Katz, A. and Quiquandon, M. (1995). Geometry of approximant structures in quasicrystals, J. Phys.: Condens. Matter 7, 9101–9125.Google Scholar
[Gri99] Grimmett, G. (1999). Percolation, 2nd ed. (Springer, Berlin).CrossRefGoogle Scholar
[GS87] Grünbaum, B. and Shephard, G.C. (1987). Tilings and Patterns (Freeman, New York).Google Scholar
[GM95] Grünbaum, F.A. and Moore, C.C. (1995). The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets, Acta Cryst. A 51, 310–323.CrossRefGoogle Scholar
[GS85] Guiaşu, S. and Shenitzer, A. (1985). The principle of maximum entropy, Math. Intelligencer 7, 42–48.CrossRefGoogle Scholar
[Gum96] Gummelt, P. (1996). Penrose tilings as coverings of congruent decagons, Geom. Dedicata 62, 1–17.CrossRefGoogle Scholar
[Gum99] Gummelt, P. (1999). Aperiodische Überdeckungen mit einem Clustertyp, PhD thesis, Univ. Greifswald (Shaker, Aachen).Google Scholar
[H-AE+09] Haji-Akbari, A., Engel, M., Keys, A.S., Zheng, X., Petschek, R.G., Palffy-Muhoray, P. and Glotzer, S.C. (2009). Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra, Nature 462, 773–777. arXiv:1012.5138.CrossRefGoogle ScholarPubMed
[Hal44] Halmos, P.R. (1944). In general a measure preserving transformation is mixing, Ann. Math. 45, 786–792.CrossRefGoogle Scholar
[HvN44] Halmos, P.R. and von Neumann, J. (1944), Operator methods in classical mechanics. II. Ann. Math. 43, 332–350.Google Scholar
[HW08] Hardy, G.H. and Wright, E.M. (2008). An Introduction to the Theory of Numbers, 6th ed., revised by Heath-Brown, D.R. and Silverman, J.H. (Oxford University Press, Oxford).Google Scholar
[HFonl] Harriss, E. and Frettlöh, D. (eds.). Tilings Encyclopedia, available at http://tilings.math.uni-bielefeld.de/.
[H-Y11] Hayashi, H., Kawachi, Y., Komatsu, K., Konda, A., Kurozoe, M., Nakano, F., Odawara, N., Onda, R., Sugio, A. and Yamauchi, M. (2011). Notes on vertex atlas of Danzer tiling, Nihonkai Math. J. 22, 49–58.Google Scholar
[HZ74] Hecht, E. and Zajac, A. (1974). Optics (Addison-Wesley, Reading, MA).Google Scholar
[Hen86] Henley, C.L. (1986). Sphere packings and local environments in Penrose tilings, Phys. Rev. B 34, 797–816.CrossRefGoogle ScholarPubMed
[Hen98] Henley, C.L. (1998). Cluster maximization, non-locality, and random tilings. In Proccedings of the 6th International Conference on Quasicrystals, Takeuchi, S. and Fujiwara, T. (eds.), pp. 27–30 (World Scientific, Singapore).Google Scholar
[Hen99] Henley, C.L. (1999). Random tiling models. In Quasicrystals: The State of the Art, 2nd ed., DiVincenzo, D.P. and Steinhardt, P.J. (eds.), pp. 459–560 (World Scientific, Singapore).Google Scholar
[Her49] Hermann, C. (1949). Kristallographie in Räumen beliebiger Dimensionszahl. I. Die Symmetrieoperationen, Acta Cryst. 2, 139–145.CrossRefGoogle Scholar
[HRB97] Hermisson, J., Richard, C. and Baake, M. (1997). A guide to the symmetry structure of quasiperiodic tiling classes, J. Phys. I France 7, 1003–1018. mp_arc/02-180.CrossRefGoogle Scholar
[Her13] Herning, J.L. (2013). Spectrum and Factors of Substitution Dynamical Systems, PhD thesis (George Washington University, Washington, DC).Google Scholar
[HR97] Hewitt, E. and Ross, K.A. (1997). Abstract Harmonic Analysis I, 2nd ed., corr. 3rd printing (Springer, New York).Google Scholar
[Hil85] Hiller, H. (1985). The crystallographic restriction in higher dimensions, Acta Cryst. A 41, 541–544.CrossRefGoogle Scholar
[Hof95a] Hof, A. (1995). On Difiraction by aperiodic structures, Commun. Math. Phys. 169, 25–43.CrossRefGoogle Scholar
[Hof95b] Hof, A. (1995). Difiraction by aperiodic structures at high temperatures. J. Phys. A: Math. Gen. 28, 57–62.CrossRefGoogle Scholar
[Hof97a] Hof, A. (1997). Difiraction by aperiodic structures. In [Moo97], pp. 239–268.Google Scholar
[Hof97b] Hof, A. (1997). On scaling in relation to singular spectra, Commun. Math. Phys. 184, 567–577.CrossRefGoogle Scholar
[HKS95] Hof, A., Knill, O. and Simon, B. (1995). Singular continuous spectrum for palindromic Schrödinger operators, Commun. Math. Phys. 174, 149–159.CrossRefGoogle Scholar
[Höf00] Höffe, M. (2000). Difiraction of the dart-rhombus random tiling, Math. Sci. Eng. A 294–296, 373–376. arXiv:math-ph/9911014.Google Scholar
[Höf01] Höffe, M. (2001). Diffraktionstheorie stochastischer Parkettierungen, PhD thesis, Univ. Tübingen (Shaker, Aachen).Google Scholar
[HB00] Höffe, M. and Baake, M. (2000). Surprises in diffuse scattering, Z. Krist. 215, 441–444. arXiv:math-ph/0004022.Google Scholar
[HKPM97] Hohneker, C., Kramer, P., Papadopolos, Z. and Moody, R.V. (1997). Canonical icosahedral quasilattices for the F-phase generated by coherent phases in physical space, J. Phys. A: Math. Gen. 30, 6493–6507.CrossRefGoogle Scholar
[HS03] Hollander, M. and Solomyak, B. (2003). Two-symbol Pisot substitutions have pure discrete spectrum, Ergod. Th. & Dynam. Syst. 23, 533–540.CrossRefGoogle Scholar
[HRS05] Holton, C., Radin, C. and Sadun, L. (2005). Conjugacies for tiling dynamical systems, Commun. Math. Phys. 254, 343–359.CrossRefGoogle Scholar
[HKPV09] Hough, J.B., Krishnapur, M., Peres, Y. and Virág, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes (AMS, Providence, RI).CrossRefGoogle Scholar
[Huc09] Huck, C. (2009). Discrete tomography of icosahedral model sets, Acta Cryst. A 65, 240–248. arXiv:0705.3005.CrossRefGoogle ScholarPubMed
[Hum92] Humphreys, J.E. (1992). Reflection Groups and Coxeter Groups, 2nd corr. printing (Cambridge University Press, Cambridge).Google Scholar
[Hur19] Hurwitz, A. (1919). Vorlesungen über die Zahlentheorie der Quaternionen (Springer, Berlin).CrossRefGoogle Scholar
[Hut81] Hutchinson, J.E. (1981). Fractals and self-similarity, Indiana Univ. Math. J. 30, 713–743.CrossRefGoogle Scholar
[Ing99] Ingersent, K. (1999). Matching rules for quasicrystalline tilings. In Quasicrystals: The State of the Art, 2nd ed., DiVincenzo, D.P. and Steinhardt, P.J. (eds.), pp. 197–224 (World Scientific, Singapore).Google Scholar
[INF85] Ishimasa, T., Nissen, H.-U. and Fukano, Y. (1985). New ordered state between crystalline and amorphous in Ni-Cr particles, Phys. Rev. Lett. 55, 511–513.CrossRefGoogle ScholarPubMed
[IK04] Iwaniec, H. and Kowalski, E. (2004). Analytic Number Theory (AMS, Providence, RI).CrossRefGoogle Scholar
[JK69] Jacobs, K. and Keane, M. (1969). 0-1-sequences of Toeplitz type, Z. Wahrscheinlichkeitsth. verw. Geb. 13, 123–131.CrossRefGoogle Scholar
[JK81] James, G. and Kerber, A. (1981). The Representation Theory of the Symmetric Group (Addison-Wesley, Reading, MA).Google Scholar
[JL01] James, G. and Liebeck, M. (2001). Representations and Characters of Groups, 2nd ed. (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[JJ77] Janner, A. and Janssen, T. (1977). Symmetry of periodically distorted crystals, Phys. Rev. B 15, 643–658.CrossRefGoogle Scholar
[Jan94] Janot, C. (1994). Quasicrystals: A Primer, 2nd ed. (Clarendon Press, Oxford).CrossRefGoogle Scholar
[JCdB07] Janssen, T., Chapuis, G. and de Boissieu, M. (2007). Aperiodic Crystals — From Modulated Phases to Quasicrystals (Clarendon Press, Oxford).CrossRefGoogle Scholar
[Jar88] Jarić, M.V. (ed.) (1988). Introduction to Quasicrystals, Aperiodicity & Order, vol. 1 (Academic Press, San Diego, CA).
[Jar89] Jarić, M.V. (ed.) (1989). Introduction to the Mathematics of Quasicrystals, Aperiodicity & Order, vol. 2 (Academic Press, San Diego, CA).
[JG89] Jarić, M.V. and Gratias, D. (eds.) (1989). Extended Icosahedral Structures, Aperiodicity & Order, vol. 3 (Academic Press, San Diego, CA).
[Jay82] Jaynes, E.T. (1982). On the rationale of maximum entropy methods, Proc. IEEE 70, 939–952.CrossRefGoogle Scholar
[JS94] Jeong, H.-C. and Steinhardt, P.J. (1994). Cluster approach for quasicrystals, Phys. Rev. Lett. 73, 1943–1946.CrossRefGoogle ScholarPubMed
[JM97] Johnson, A. and Madden, K. (1997). Putting the pieces together: Understanding Robinson's nonperiodic tilings, College Math. J. 28, 172–181.CrossRefGoogle Scholar
[Jul10] Julien, A. (2010). Complexity and cohomology for cut-and-projection tilings, Ergodic Th. & Dynam. Syst. 30, 489–523. arXiv:0804.0145.Google Scholar
[Kak72] Kakutani, S. (1972). Strictly ergodic symbolic dynamical systems. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Le Cam, L.M., Neyman, J. and Scott, E.L. (eds.), pp. 319–326 (University of California Press, Berkeley).Google Scholar
[Kal94] Kalugin, P.A. (1994). The square-triangle random-tiling model in the thermodynamic limit, J. Phys. A: Math. Gen. 27, 3599–3614.CrossRefGoogle Scholar
[Kal05] Kalugin, P. (2005). Cohomology of quasiperiodic patterns and matching rules, J. Phys. A: Math. Gen. 38, 3115–3132.CrossRefGoogle Scholar
[Kar96] Kari, J. (1996). A small aperiodic set of Wang tiles, Discr. Math. 160, 259–264.CrossRefGoogle Scholar
[KT75] Karlin, S. and Taylor, H.M. (1975). A First Course in Stochastic Processes, 2nd ed. (Academic Press, San Diego, CA).Google Scholar
[Kar91] Karr, A.F. (1991). Point Processes and their Statistical Inference, 2nd ed. (Dekker, New York).Google Scholar
[Kas63] Kasteleyn, P.W. (1963). Dimer statistics and phase transitions, J. Math. Phys. 4, 287–293.CrossRefGoogle Scholar
[KH95] Katok, A. and Hasselblatt, B. (1995). Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[Kat88] Katz, A. (1988). Theory of matching rules for the 3-dimensional Penrose tilings, Commun. Math. Phys. 118, 263–288.CrossRefGoogle Scholar
[Kat95] Katz, A. (1995). Matching rules and quasiperiodicity: The octagonal tilings. In [AG95], pp. 141–189.Google Scholar
[KG94] Katz, A. and Gratias, D. (1994). Tilings and Quasicrystals. In Lectures on Quasicrystals, Hippert, F. and Gratias, D. (eds.), pp. 187–264 (Les Editions des Physique, Les Ulis).Google Scholar
[Kat04] Katznelson, Y. (2004). An Introduction to Harmonic Analysis, 3rd ed. (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[Kea68] Keane, M. (1968). Generalized Morse sequences, Z. Wahrscheinlichkeitsth. verw. Geb. 10, 335–353.CrossRefGoogle Scholar
[KP06] Keane, M. and Petersen, K. (2006). Easy and nearly simultaneous proofs of the ergodic theorem and maximal ergodic theorem. In Dynamics & Stochastics: Festschrift in honor of M.S. Keane, Denteneer, D., den Hollander, F. and Verbitskiy, E. (eds.), pp. 248–251 (Institute of Mathematical Statistics, Beachwood, OH).Google Scholar
[Kel95] Kellendonk, J. (1995). Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys. 7, 1133–1180. arXiv:cond-mat/9403065.CrossRefGoogle Scholar
[KS12] Kellendonk, J. and Sadun, L. (2012). Meyer sets, topological eigenvalues, and Cantor fiber bundles, Preprint arXiv:1211.2250.Google Scholar
[Kel98] Keller, G. (1998). Equilibrium States in Ergodic Theory (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[Ken97] Kenyon, R. (1997). Local statistics of lattice dimers, Ann. Inst. H. Poincaré B 33, 591–618. arXiv:math.CO/0105054.Google Scholar
[Ken00] Kenyon, R. (2000). The planar dimer model with boundary: A survey. In [BM00b], pp. 307–328.Google Scholar
[KS10] Kenyon, R. and Solomyak, B. (2010). On the characterization of expansion maps for self-afine tilings, Discr. Comput. Geom. 43, 577–593. arXiv:0801.1993.CrossRefGoogle Scholar
[Kep19] Kepler, J. (1619). Harmonices Mundi Libri V (J. Planck for G. Tampach, Linz).CrossRefGoogle Scholar
[Khi57] Khinchin, A.I. (1957). Mathematical Foundations of Information Theory (Dover, New York).Google Scholar
[KS09] Killip, R.and, M.Stoiciu, M. (2009). Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles, Duke Math. J. 146, 361–399. arXiv:math-ph/0608002.CrossRefGoogle Scholar
[KET12] Kiselev, A., Engel, M. and Trebin, H.-R. (2012). Confirmation of the random tiling hypothesis for a decagonal quasicrystal, Phys. Rev. Lett. 109, 225502:1–4. arXiv:1210.4227.CrossRefGoogle ScholarPubMed
[Kit98] Kitchens, B.P. (1998). Symbolic Dynamics (Springer, Berlin).CrossRefGoogle Scholar
[Kle84] Klein, F. (1884). Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade (Teubner, Leipzig).Google Scholar
[KSB93] Klitzing, R., Schlottman, M. and Baake, M. (1993). Perfect matching rules for undecorated triangular tilings with 10-, 12-, and 8-fold symmetry, Int. J. Mod. Phys. B 7, 1455–1473.CrossRefGoogle Scholar
[Kni98] Knill, O. (1998). Singular continuous spectrum and quantitative rates of weak mixing, Discr. Cont. Dynam. Syst. 4, 33–42.Google Scholar
[Kob96] Koblitz, N. (1996). p-adic Numbers, p-adic Analysis, and Zeta-Functions, corr. 2nd ed. (Springer, New York).Google Scholar
[KR91] Koecher, M. and Remmert, R. (1991). Hamilton's quaternions. In Numbers, Ebbinghaus, H.-D.et al. (eds.), pp. 189–220 (Springer, New York).Google Scholar
[Kol65] Kolakoski, W. (1965). Self-generating runs; problem 5304, Amer. Math. Monthly 72, 674.Google Scholar
[Koo31] Koopman, B.O. (1931). Hamiltonian systems and transformations in Hilbert space, Proc. Nat. Acad. Sci. USA 17, 315–318.CrossRefGoogle Scholar
[Kow38] Kowalewski, G. (1938). Der Keplersche Körper und andere Bauspiele (Koehlers Antiquarium, Leipzig).Google Scholar
[Kra82] Kramer, P. (1982). Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells, Acta Cryst. A 38, 257–264.CrossRefGoogle Scholar
[Kra87a] Kramer, P. (1987). Atomic order in quasicrystals is supported by several unit cells, Mod. Phys. Lett. B 1, 7–18.CrossRefGoogle Scholar
[Kra87b] Kramer, P. (1987). Continuous rotation from cubic to icosahedral order, Acta Cryst. A 43, 486–489.CrossRefGoogle Scholar
[Kra88] Kramer, P. (1988). Space-group theory for a nonperiodic icosahedral quasilattice, J. Math. Phys. 29, 516–524.CrossRefGoogle Scholar
[KH89] Kramer, P. and Haase, R.W. (1989). Group theory of icosahedral quasicrystals. In [Jar89], pp. 81–146.Google Scholar
[KN84] Kramer, P. and Neri, R. (1984) On periodic and non-periodic space fillings of Em obtained by projectionActa Cryst. A 40, 580–587 and Acta Cryst. A 41 (1985), 619 (erratum).CrossRefGoogle Scholar
[KP03] Kramer, P. and Papadopolos, Z. (eds.) (2003). Coverings of Discrete Quasiperiodic Sets (Springer, Berlin).CrossRef
[KPSZ94] Kramer, P., Papadopolos, Z., Schlottmann, M. and Zeidler, D. (1994). Projection of the Danzer tiling, J. Phys. A: Math. Gen. 27, 4505–4517.CrossRefGoogle Scholar
[KS89] Kramer, P. and Schlottmann, M. (1989). Dualisation of Voronoi domains and Klotz construction: A general method for the generation of proper space fillings, J. Phys. A: Math. Gen. 22, L1097–L1102.CrossRefGoogle Scholar
[KN74] Kuipers, L. and Niederreiter, H. (1974). Uniform Distribution of Sequences (Wiley, New York). Reprint (Dover, New York, 2006).Google Scholar
[Kül03a] Külske, C. (2003). Universal bounds on the selfaveraging of random Difiraction measures, Probab. Th. Relat. Fields 126, 29–50. arXiv:math-ph/0109005.CrossRefGoogle Scholar
[Kül03b] Külske, C. (2003). Concentration inequalities for functions of Gibbs fields with application to Difiraction and random Gibbs measures, Commun. Math. Phys. 239, 29–51.CrossRefGoogle Scholar
[Kwa11] Kwapisz, J. (2011). Rigidity and mapping class group for abstract tiling spaces, Ergodic Th. & Dynam. Syst. 31, 1745–1783.Google Scholar
[Lag96] Lagarias, J.C. (1996). Meyer's concept of quasicrystal and quasiregular sets, Commun. Math. Phys. 179, 365–376.CrossRefGoogle Scholar
[Lag99] Lagarias, J.C. (1999). Geometric models for quasicrystals I. Delone sets of finite type, Discr. Comput. Geom. 21, 161–191.CrossRefGoogle Scholar
[LP02] Lagarias, J.C. and Pleasants, P.A.B. (2002). Local complexity of Delone sets and crystallinity, Can. Math. Bull. 48, 634–652. arXiv:math.MG/0105088.Google Scholar
[LP03] Lagarias, J.C. and Pleasants, P.A.B. (2003). Repetitive Delone sets and quasicrystals, Ergod. Th. & Dynam. Syst. 23, 831–867. arXiv:math.DS/9909033.CrossRefGoogle Scholar
[LW96] Lagarias, J.C. and Wang, Y. (1996). Self-afine tiles in ℝn, Adv. Math. 121, 21–49.CrossRefGoogle Scholar
[LW03] Lagarias, J.C. and Wang, Y. (2003). Substitution Delone sets, Discr. Comput. Geom. 29, 175–209. arXiv:math.MG/0110222.CrossRefGoogle Scholar
[LZ12] Lagarias, J.C. and Zong, C. (2012). Mysteries in packing regular tetrahedra, Notices Amer. Math. Soc. 59, 1540–1549.CrossRefGoogle Scholar
[LB88] Lançon, F. and Billard, L. (1988). Two-dimensional system with a quasicrystalline ground state, J. Physique (France) 49, 249–256.CrossRefGoogle Scholar
[Lan93] Lang, S. (1993). Real and Functional Analysis, 3rd ed. (Springer, New York).CrossRefGoogle Scholar
[Lan02] Lang, S. (2002). Algebra, rev. 3rd ed. (Springer, New York).CrossRefGoogle Scholar
[Le95] Le, T.T.Q. (1995). Local rules for pentagonal quasi-crystals, Discr. Comput. Geom. 14, 31–70.Google Scholar
[Le97] Le, T.T.Q. (1997). Local rules for quasiperiodic tilings. In [Moo97], pp. 331–366.Google Scholar
[Led78] Ledrappier, F. (1978). Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B 287, A561–A563.Google Scholar
[LM01] Lee, J.-Y. and Moody, R.V. (2001). Lattice substitution systems and model sets, Discr. Comput. Geom. 25, 173–201. arXiv:math.MG/0002019.CrossRefGoogle Scholar
[LM08] Lee, J.-Y. and Moody, R.V. (2008). Deforming Meyer sets, Europ. J. Combin. 29, 1919–1924. arXiv:0910.4446.CrossRefGoogle Scholar
[LM13] Lee, J.-Y. and Moody, R.V. (2013). Taylor-Socolar hexagonal tilings as model sets, Symmetry 5, 1–46. arXiv:1207.6237.Google Scholar
[LMS02] Lee, J.-Y., Moody, R.V. and Solomyak, B. (2002). Pure point dynamical and Difiraction spectra, Ann. H. Poincaré 3, 1003–1018. arXiv:0910.4809.CrossRefGoogle Scholar
[Len02] Lenz, D. (2002). Uniform ergodic theorems on subshifts over a finite alphabet, Ergod. Th. & Dynam. Syst. 22, 245–255. arXiv:math.DS/0005067.CrossRefGoogle Scholar
[Len09] Lenz, D. (2009). Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Commun. Math. Phys. 287, 225–258. arXiv:math-ph/0608026.CrossRefGoogle Scholar
[LM09] Lenz, D. and Moody, R.V. (2009). Extinctions and correlations for uniformly discrete point processes with pure point dynamical spectra. Commun. Math. Phys. 289, 907–923. arXiv:0902.0567.CrossRefGoogle Scholar
[LM11] Lenz, D. and Moody, R.V. (2011). Stationary processes with pure point diffraction, Preprint arXiv:1111.3617.Google Scholar
[LR07] Lenz, D. and Richard, C. (2007). Pure point Difiraction and cut and project schemes for measures: The smooth case, Math. Z. 256, 347–378. arXiv:math.DS/0603453.CrossRefGoogle Scholar
[LS84] Levine, D. and Steinhardt, P.J. (1984). Quasicrystals: A new class of ordered structures, Phys. Rev. Lett. 53, 2477–2480.CrossRefGoogle Scholar
[Lev88] Levitov, L.S. (1988). Local rules for quasicrystals, Commun. Math. Phys. 119, 627–666.CrossRefGoogle Scholar
[Lif96] Lifshitz, R. (1996). The symmetry of quasiperiodic crystals, Physica A 232, 633–647.CrossRefGoogle Scholar
[Lif 97] Lifshitz, R. (1997). Theory of color symmetry for periodic and quasiperiodic crystals, Rev. Mod. Phys. 69, 1181–1218.CrossRefGoogle Scholar
[LD07] Lifshitz, R. and Diamant, H. (2007). Soft quasicrystals — why are they stable?Philos. Mag. 87, 3021–3030. arXiv:cond-mat/0611115.CrossRefGoogle Scholar
[Lin04] Lind, D. (2004). Multi-dimensional symbolic dynamics, Proc. Sympos. Appl. Math. 60, 61–79.Google Scholar
[LM95] Lind, D.A. and Marcus, B. (1995). An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[LSW90] Lind, D.A., Schmidt, K. and Ward, T. (1990). Mahler measure and entropy for commuting automorphisms of compact groups, Inv. Math. 101, 593–629.CrossRefGoogle Scholar
[Loo11] Loomis, L.H. (2011). Introduction to Abstract Harmonic Analysis, reprint (Dover, New York).Google Scholar
[LRK00] Lord, E.A., Ranganathan, S. and Kulkarni, U.D. (2000). Tilings, coverings, clusters and quasicrystals, Current Sci. 78, 64–72.Google Scholar
[Lot02] Lothaire, M. (2002). Algebraic Combinatorics on Words (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[LGJJ93] Luck, J.M., Godrèche, C., Janner, A. and Janssen, T. (1993). The nature of the atomic surfaces of quasiperiodic self-similar structures, J. Phys. A: Math. Gen. 26, 1951–1999.CrossRefGoogle Scholar
[Lüc88] Lück, R. (1988). Description of imperfections in the icosahedral Penrose tiling. In Quasicrystalline Materials, Janot, Ch. and Dubois, J.M. (eds.), pp. 308–317 (World Scientific, Singapore).Google Scholar
[Lüc93] Lück, R. (1993). Basic ideas of Ammann bar grids, Int. J. Mod. Phys. B 7, 1437–1453.CrossRefGoogle Scholar
[Lüc00] Lück, R. (2000). Dürer-Kepler-Penrose, the development of pentagonal tilings, Mat. Sci. Eng. A 294–296, 263–267.Google Scholar
[LP92] Lunnon, W.F. and Pleasants, P.A.B. (1992). Characterization of two-distance sequences, J. Austral. Math. Soc. (Series A) 53, 198–218.CrossRefGoogle Scholar
[LAT04] Luo, J., Akiyama, S. and Thuswaldner, J.M. (2004). On the boundary connectedness of connected tiles, Math. Proc. Cambridge Philos. Soc. 137, 397–410.CrossRefGoogle Scholar
[Mac81] Mackay, A.L. (1981). De nive quinquangula: On the pentagonal snowflake, Sov. Phys. Cryst. 26, 517–522.Google Scholar
[Mac82] Mackay, A.L. (1982). Crystallography and the Penrose pattern, Physica A 114, 609–613.CrossRefGoogle Scholar
[MKS76] Magnus, W., Karrass, A. and Solitar, D. (1976). Combinatorial Group Theory, rev. ed. (Dover, New York).Google Scholar
[Mah27] Mahler, K. (1927). The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions. Part II: On the translation properties of a simple class of arithmetical functions, J. Math. Massachusetts 6, 158–163.Google Scholar
[Mak92] Makovicky, E. (1992). 800-year-old pentagonal tiling from Marāgha, Iran, and the new varieties of new aperiodic tiling it inspired. In Fivefold Symmetry, Hargittai, I. (ed.), pp. 67–86 (World Scientific, Singapore).Google Scholar
[Mat75] Matheron, G. (1975). Random Sets and Integral Geometry (Wiley, New York).Google Scholar
[Mat86] Matheron, G. (1986). Le covariogramme géometrique des compacts convexes de ℝ2, Technical Report 2/86 (Centre de Géostatistique, École des Mines de Paris).Google Scholar
[MN83] Mathew, J. and Nadkarni, M.G. (1983). A measure preserving transformation whose spectrum has Lebesgue component of multiplicity two, Bull. London Math. Soc. 16, 402–406.Google Scholar
[Mat10] Matzutt, K. (2010). Difiraction of point sets with structural disorder, PhD thesis (Univ. Bielefeld).Google Scholar
[McM98] McMullen, C.T. (1998). Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal. 8, 304–314.CrossRefGoogle Scholar
[MS02] McMullen, P. and Schulte, E. (2002). Abstract Regular Polytopes (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[Meh04] Mehta, M.L. (2004). Random Matrices, 3rd ed. (Elsevier, Amsterdam).Google Scholar
[MRW87] Mermin, N.D., Rokhsar, D.S. and Wright, D.C. (1987). Beware of 46-fold symmetry: The Classification of two-dimensional quasicrystallographic lattices, Phys. Rev. Lett. 58, 2099–2101.CrossRefGoogle ScholarPubMed
[Mey70] Meyer, Y. (1970). Nombres de Pisot, Nombres de Salem et Analyse Harmonique, LNM 117 (Springer, Berlin).CrossRefGoogle Scholar
[Mey72] Meyer, Y. (1972). Algebraic Numbers and Harmonic Analysis (North Holland, Amsterdam).Google Scholar
[Mey12] Meyer, Y. (2012). Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling, African Diaspora J. Math. 13, 1–45.Google Scholar
[Miȩ93] Miȩkisz, J. (1993). Quasicrystals: Microscopic Models on Nonperiodic Structures (Leuven University Press, Leuven).Google Scholar
[MvE90] Miȩkisz, J. and van Enter, A.C.D. (1990). Breaking of periodicity at positive temperatures, Commun. Math. Phys. 134, 647–651.Google Scholar
[MSR+10] Mikhael, J., Schmiedeberg, M., Rausch, S., Roth, J., Stark, H. and Bechinger, C. (2010), Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields, PNAS 107, 7214–7218. arXiv:1005.2120.CrossRefGoogle ScholarPubMed
[Moo97] Moody, R.V. (ed.) (1997). The Mathematics of Long-Range Aperiodic Order, NATO ASI Series C 489 (Kluwer, Dordrecht).CrossRef
[Moo97a] Moody, R.V. (1997). Meyer sets and their duals. In [Moo97], pp. 403–441.Google Scholar
[Moo00] Moody, R.V. (2000). Model sets: A Survey. In From Quasicrystals to More Complex Systems, Axel, F., Dénoyer, F. and Gazeau, J.P. (eds.), pp. 145–166 (EDP Sciences, Les Ulis, and Springer, Berlin). arXiv:math.MG/0002020.Google Scholar
[Moo02] Moody, R.V. (2002). Uniform distribution in model sets, Can. Math. Bull. 45, 123–130.CrossRefGoogle Scholar
[MP92] Moody, R.V. and Patera, J. (1992). Voronoi and Delone cells of root lattices: Classification of their faces and facets by Coxeter–Dynkin diagram, J. Phys. A: Math. Gen. 25, 5089–5134.CrossRefGoogle Scholar
[MP93] Moody, R.V. and Patera, J. (1993). Quasicrystals and icosians, J. Phys. A: Math. Gen. 26, 2829–2853.CrossRefGoogle Scholar
[MP96] Moody, R.V. and Patera, J. (1996). Dynamical generation of quasicrystals, Lett. Math. Phys. 36, 291–300.CrossRefGoogle Scholar
[MW94] Moody, R.V. and Weiss, A. (1994). On shelling E8 quasicrystals, J. Number Th. 47, 405–412.CrossRefGoogle Scholar
[MH38] Morse, M. and Hedlund, G.A. (1938). Symbolic dynamics, Amer. J. Math. 60, 815–866.CrossRefGoogle Scholar
[MH40] Morse, M. and Hedlund, G.A. (1940). Symbolic dynamics II, Amer. J. Math. 62, 1–42.CrossRefGoogle Scholar
[Mou10] Moustafa, H. (2010). PV cohomology of the pinwheel tilings, their integer group of coinvariants and gap-labeling, Commun. Math. Phys. 298, 369–405. arXiv:0906.2107.CrossRefGoogle Scholar
[Moz89] Mozes, S. (1989). Tilings, substitution systems and dynamical systems generated by them, J. d'Analyse Math. 53, 139–186.CrossRefGoogle Scholar
[MR13] Müller, P. and Richard, C. (2013). Ergodic properties of randomly coloured point sets, Can. J. Math. 65, 349–402. arXiv:1005.4884.CrossRefGoogle Scholar
[Nad95] Nadkarni, M.G. (1995). Basic Ergodic Theory, 2nd ed. (Birkhäuser, Basel).CrossRefGoogle Scholar
[Nad98] Nadkarni, M.G. (1998). Spectral Theory of Dynamical Systems (Birkhäuser, Basel).Google Scholar
[NM87] Niizeki, N. and Mitani, H. (1987). Two-dimensional dodecagonal quasilattices, J. Phys. A: Math. Gen. 20, L405–L410.CrossRefGoogle Scholar
[NHL13] Nikola, N., Hexner, D. and Levine, D. (2013). Entropic commensurate- incommensurate transition, Phys. Rev. Lett. 110, 125701. arXiv:1211.6972.CrossRefGoogle ScholarPubMed
[Nil12a] Nilsson, J. (2012). On the entropy of a family of random substitution systems, Monatsh. Math. 166, 1–15. arXiv:1001.3513 and arXiv:1103.4777.Google Scholar
[Nil12b] Nilsson, J. (2012). A space efficient algorithm for the calculation of the digit distribution in the Kolakoski sequence, J. Integer Sequences 15, 12.6.7: 1–13. arXiv:1110.4228.Google Scholar
[ND96] Nischke, K.-P. and Danzer, L. (1996). A construction of inflation rules based on n-fold symmetry, Discr. Comput. Geom. 15, 221–236.CrossRefGoogle Scholar
[Off08] Offner, C.D. (2008). Repetitions of words and the Thue-Morse sequence, unpublished notes; available from http://www.cs.umb.edu/~offner/Google Scholar
[Old39] Oldenburger, R. (1939). Exponent trajectories in symbolic dynamics, Trans. Amer. Math. Soc. 46, 453–466.CrossRefGoogle Scholar
[O'Mea73] O'Meara, O.T. (1973). Introduction to Quadratic Forms, 3rd. corr. printing (Springer, Berlin); reprint (Springer, Berlin, 2000).CrossRefGoogle Scholar
[OH93] Oxborrow, M. and Henley, C.L. (1993). Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals, Phys. Rev. B 48, 6966–6998.CrossRefGoogle ScholarPubMed
[Oxt52] Oxtoby, J.C. (1952). Ergodic sets, Bull. Amer. Math. Soc. 58, 116–136.CrossRefGoogle Scholar
[PHK00] Papadopolos, Z., Hohneker, C. and Kramer, P. (2000). Tiles-inflation rules for the class of canonical tilings T*(2F) derived by the projection method. Discr. Math. 221, 101–112.CrossRefGoogle Scholar
[PKK97] Papadopolos, Z., Klitzing, R. and Kramer, P. (1997). Quasiperiodic icosahedral tilings from the six-dimensional bcc lattice, J. Phys. A: Math. Gen. 30, L143–L147.CrossRefGoogle Scholar
[Par67] Parthasarathy, K.R. (1967). Probability Measures on Metric Spaces (Academic Press, New York).CrossRefGoogle Scholar
[Pat97] Patera, J. (1997). Noncrystallographic root systems and quasicrystals. In [Moo97], pp. 443–465.Google Scholar
[Pat98] Patera, J. (ed.) (1998). Quasicrystals and Discrete Geometry, Fields Institute Monographs, vol. 10 (AMS, Providence, RI).
[Patt44] Patterson, A.L. (1944). Ambiguities in the X-ray analysis of crystal structures, Phys. Rev. 65, 195–201.CrossRefGoogle Scholar
[PK87] Pavlovitch, A. and Kléman, M. (1987). Generalised 2D Penrose tilings: Structural properties, J. Phys. A: Math. Gen. 20, 687–702.CrossRefGoogle Scholar
[Ped95] Pedersen, G.K. (1995). Analysis Now, rev. printing (Springer, New York).Google Scholar
[PP58a] Penrose, L.S. and Penrose, R. (1958). Impossible objects: A special type of visual illusion, Brit. J. Psychol. 49, 31–33; reprinted in Roger Penrose Collected Works, vol. 1, pp. 355–358 (2011, Oxford University Press, Oxford).CrossRefGoogle ScholarPubMed
[PP58b] Penrose, L.S. and Penrose, R. (1958). Puzzles for Christmas, New Scientist (Dec. 25) 1580–1581 and 1597–1598; reprinted in Roger Penrose Collected Works, vol. 1, pp. 359–363 (2011, Oxford University Press, Oxford).Google Scholar
[Pen74] Penrose, R. (1974). The rôle of aesthetics in pure and applied mathematical research, Bull. Inst. Math. Appl. 10, 266–271.Google Scholar
[Pen78] Penrose, R. (1978). Pentaplexity, Eureka 39, 16–22; reprinted in Math. Intelligencer 2 (1979), 32–38.Google Scholar
[Pen97] Penrose, R. (1997). Remarks on tiling: Details of a (1 + ε + ε2)-aperiodic set. In [Moo97], pp. 467–497; a supplement appeared in the Twistor Newsletter (TN 41 (1996) 37, TN 42 (1997) 25–26 and TN 43 (1998) 34); all reprinted in Roger Penrose Collected Works, vol. 6 (2011, Oxford University Press, Oxford).Google Scholar
[Per54] Perron, O. (1954). Die Lehre von den Kettenbrüchen, 3rd ed. (Teubner, Stuttgart).Google Scholar
[Pet83] Petersen, K. (1983). Ergodic Theory (Cambridge University Press, Cambridge).CrossRefGoogle Scholar
[Pin02] Pinsky, M.A. (2002). Introduction to Fourier Analysis and Wavelets (Brooks/Cole, Pacific Grove, CA).Google Scholar
[Ple00] Pleasants, P.A.B. (2000). Designer quasicrystals: Cut-and-project sets with pre-assigned properties. In [BM00b], pp. 95–141.Google Scholar
[PH13] Pleasants, P.A.B. and Huck, C. (2013). Entropy and Difiraction of the k-free points in n-dimensional lattices, Discr. Comput. Geom. 50, in press. arXiv:1112.1629.CrossRefGoogle Scholar
[Pow11] Power, S.C. (2011). Crystal frameworks, matrix-valued functions and rigidity operators, Preprint arXiv:1111.2943.Google Scholar
[Pra94] Prasolov, V.V. (1994). Problems and Theorems in Linear Algebra (AMS, Providence, RI).CrossRefGoogle Scholar
[PF02] Pytheas Fogg, N. (2002). Substitutions in Dynamics, Arithmetics and Combinatorics, LNM 1794 (Springer, Berlin).CrossRefGoogle Scholar
[Que95] Queffélec, M. (1995). Spectral study of automatic and substitutive sequences. In [AG95], pp. 369–414.Google Scholar
[Que10] Queffélec, M. (2010). Substitution Dynamical Systems — Spectral Analysis, LNM 1294, 2nd ed. (Springer, Berlin).CrossRefGoogle Scholar
[Rad87] Radin, C. (1987). Low temperature and the origin of crystalline symmetry, Int. J. Mod. Phys. B 1, 1157–1191.CrossRefGoogle Scholar
[Rad99] Radin, C. (1999). Miles of Tiles (AMS, Providence, RI).CrossRefGoogle Scholar
[RS98] Radin, C. and Sadun, L. (1998). Subgroups of SO(3) associated with tilings, J. Algebra 202, 611–633. arXiv:math.GR/9601202.CrossRefGoogle Scholar
[RS99] Radin, C. and Sadun, L. (1999). On 2-generator subgroups of SO(3), Trans. Amer. Math. Soc. 351, 4469–4480.CrossRefGoogle Scholar
[RW92] Radin, C. and Wolff, M. (1992). Space tilings and local isomorphism, Geom. Dedicata 42, 355–360.CrossRefGoogle Scholar
[RS80] Reed, M. and Simon, B. (1980). Methods of Modern Mathematical Physics I: Functional Analysis, 2nd ed. (Academic Press, San Diego, CA).Google Scholar
[RG03] Reichert, M. and Gähler, F. (2003). Cluster model of decagonal tilings, Phys. Rev. B 68, 214202.CrossRefGoogle Scholar
[Rei03] Reiner, I. (2003). Maximal Orders, 2nd ed. (Clarendon Press, Oxford).Google Scholar
[ReSt00] Reiter, H. and Stegeman, J.D. (2000). Classical Harmonic Analysis and Locally Compact Groups (Clarendon Press, Oxford).Google Scholar
[Rho05] Rhoads, G.C. (2005). Planar tilings by polyominoes, polyhexes, and polyiamonds, J. Comput. Appl. Math. 174, 329–353.CrossRefGoogle Scholar
[Ric99a] Richard, C. (1999). An alternative view on random tilings, J. Phys. A: Math. Gen. 32, 8823–8829. arXiv:cond-mat/9907262.CrossRefGoogle Scholar
[Ric99b] Richard, C. (1999). Statistische Physik stochastischer Parkettierungen, PhD thesis, Univ. Tübingen (UFO Verlag, Allensbach).Google Scholar
[Ric03] Richard, C. (2003). Dense Dirac combs in Euclidean space with pure point Difiraction, J. Math. Phys. 44, 4436–4449. arXiv:math-ph/0302049.CrossRefGoogle Scholar
[RHHB98] Richard, C., Höffe, M., Hermisson, J. and Baake, M. (1998). Random tilings — concepts and examples, J. Phys. A: Math. Gen. 31, 6385–6408. arXiv:cond- mat/9712267.CrossRefGoogle Scholar
[Rob96] Robinson, E.A. (1996). The dynamical properties of Penrose tilings, Trans. Amer. Math. Soc. 348, 4447–4464.CrossRefGoogle Scholar
[Rob99] Robinson, E.A. (1999). On the table and the chair, Indag. Math. 10, 581–599.CrossRefGoogle Scholar
[Rob04] Robinson, E.A. (2004). Symbolic dynamics and tilings of ℝd, Proc. Sympos. Appl. Math. 60, 81–119.Google Scholar
[Rob71] Robinson, R.M. (1971). Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12, 177–209.CrossRefGoogle Scholar
[RMW87] Rokhsar, D.S., Mermin, N.D. and Wright, D.C. (1987). Rudimentary quasicrys-tallography: The icosahedral and decagonal reciprocal lattices, Phys. Rev. B 35, 5487–5495.CrossRefGoogle ScholarPubMed
[Rot93] Roth, J. (1993). The equivalence of two face-centered icosahedral tilings with respect to local derivability, J. Phys. A: Math. Gen. 26, 1455–1461.CrossRefGoogle Scholar
[Rud59] Rudin, W. (1959). Some theorems on Fourier coefficients, Proc. Amer. Math. Soc. 10, 855–859.CrossRefGoogle Scholar
[Rud62] Rudin, W. (1962). Fourier Analysis on Groups (Wiley, New York).Google Scholar
[Rud87] Rudin, W. (1987). Real and Complex Analysis, 3rd ed. (McGraw Hill, New York).Google Scholar
[Rud90] Rudolph, D.J. (1990). ×2 and ×3 invariant measures and entropy, Ergod. Th. & Dynam. Syst. 10, 395–406.CrossRefGoogle Scholar
[Sad98] Sadun, L. (1998). Some generalizations of the pinwheel tiling, Discr. Comput. Geom. 20, 79–110.CrossRefGoogle Scholar
[Sad08] Sadun, L. (2008). Topology of Tiling Spaces (AMS, Providence, RI).CrossRefGoogle Scholar
[SW03] Sadun, L. and Williams, R.F. (2003). Tiling spaces are Cantor set fibre bundles, Ergod. Th. & Dynam. Syst. 23, 307–316.CrossRefGoogle Scholar
[Sal63] Salem, R. (1963). Algebraic Numbers and Fourier Analysis (D. C. Heath and Company, Boston, MA).Google Scholar
[Sar11] Sarnak, P. (2011). Three lectures on the Möbius function randomness and dynamics (Lecture 1), unpublished notes, available at http://publications.ias.edu/sarnak/paper/512.Google Scholar
[Sas86] Sasisekharan, V. (1986). A new method for generation of quasi-periodic structures with n fold axes: Application to five and seven folds, Pramana J. Phys. 26, L283–L293.CrossRefGoogle Scholar
[SW99] Schaefer, H.H. and Wolff, M.P. (1999). Topological Vector Spaces, 2nd ed. (Springer, New York).CrossRefGoogle Scholar
[Schl93a] Schlottmann, M. (1993). Periodic and quasi-periodic Laguerre tilings, Int. J. Mod. Phys. B 6–7, 1351–1363.Google Scholar
[Schl93b] Schlottmann, M. (1993). Geometrische Eigenschaften quasiperiodischer Strukturen, PhD thesis (Univ. Tübingen).Google Scholar
[Schl98] Schlottmann, M. (1998). Cut-and-project sets in locally compact Abelian groups. In [Pat98], pp. 247–264.Google Scholar
[Schl00] Schlottmann, M. (2000). Generalised model sets and dynamical systems. In [BM00b], pp. 143–159.Google Scholar
[Schm95] Schmidt, K. (1995). Dynamical Systems of Algebraic Origin (Birkhäuser, Basel).Google Scholar
[SV09] Schmidt, K. and Verbitskiy, E. (2009). Abelian sandpiles and the harmonic model, Commun. Math. Phys. 292, 721–759. arXiv:0901.3124.CrossRefGoogle Scholar
[Schw98] Schwartz, L. (1998). Théorie des Distributions, 3rd ed. (Hermann, Paris).Google Scholar
[Schw80] Schwarzenberger, R.L.E. (1980). N-Dimensional Crystallography (Pitman, London).Google Scholar
[Sen95] Senechal, M. (1995). Quasicrystals and Geometry (Cambridge University Press, Cambridge).Google Scholar
[Sen04] Senechal, M. (2004). The mysterious Mr. Ammann, Math. Intelligencer 26, 10–21.CrossRefGoogle Scholar
[Sen06] Seneta, E. (2006). Non-negative Matrices and Markov Chains, 2nd rev. ed. (Springer, New York).Google Scholar
[SAH06] Shannon, A.G., Anderson, P.G. and Horadam, A.F. (2006). Properties of Cordonnier, Perrin and van der Laan numbers, Int. J. Math. Educ. Sci. Techn. 37, 825–831.CrossRefGoogle Scholar
[Sha51] Shapiro, H.S. (1951). Extremal Problems for Polynomials and Power Series, Masters thesis (MIT, Boston).Google Scholar
[SBGC84] Shechtman, D., Blech, I., Gratias, D. and Cahn, J.W. (1984). Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53, 1951–1953.CrossRefGoogle Scholar
[Sie03] Siegel, A. (2003). Représentation des systèmes dynamiques substitutifs non unimodulaires, Ergodic Th. & Dynam. Syst. 23, 1247–1273.Google Scholar
[ST10] Siegel, A. and Thuswaldner, J.M. (2010). Topological Properties of Rauzy Fractals, Mém. Soc. Math. France 102 (Société Mathématiques de France, Paris).Google Scholar
[Sie44] Siegel, C.L. (1944). Algebraic numbers whose conjugates lie in the unit circle, Duke Math. J. 11, 597–602.CrossRefGoogle Scholar
[Sim95] Simon, B. (1995). Operators with singular continuous spectrum. I. General operators, Ann. Math. 141, 131–145.CrossRefGoogle Scholar
[Sin02] Sing, B. (2002). Spektrale Eigenschaften der Kolakoski-Sequenzen, Diploma thesis (Univ. Tübingen).Google Scholar
[Sin03] Sing, B. (2003). Kolakoski-(2m, 2n) are limit-periodic model sets, J. Math. Phys. 44, 899–912. arXiv:math-ph/0207037.CrossRefGoogle Scholar
[Sin06] Sing, B. (2006). Pisot Substitutions and Beyond, PhD thesis (Univ. Bielefeld).Google Scholar
[Sin11] Sing, B. (2011). More Kolakoski sequences, Integers 11B, A14: 1–17. arXiv:1009.4061.Google Scholar
[SW06] Sing, B. and Welberry, T.R. (2006). Deformed model sets and distorted Penrose tilings, Z. Krist. 221, 621–634. mp_arc/06-199.Google Scholar
[Sin85] Singh, P. (1985). The so-called Fibonacci numbers in ancient and medieval India, Hist. Math. 12, 229–244.CrossRefGoogle Scholar
[SS02] Sirvent, V.F. and Solomyak, B. (2002). Pure discrete spectrum for one-dimensional substitution systems of Pisot type, Can. Math. Bull. 45, 697–710.CrossRefGoogle Scholar
[SW02] Sirvent, V.F. and Wang, Y. (2002). Self-a?ne tiling via substitution dynamical systems and Rauzy fractals, Pacific J. Math. 206, 465–485.CrossRefGoogle Scholar
[Sla81] Slawny, J. (1981). Ergodic properties of equilibrium states, Commun. Math. Phys. 80, 477–483.CrossRefGoogle Scholar
[Sloane] Sloane, N.J.A. (ed.). The On-Line Encyclopedia of Integer Sequences, available at http://oeis.org/.CrossRef
[Soc89] Socolar, J.E.S. (1989). Simple octagonal and dodecagonal quasicrystals, Phys. Rev. B 39, 10519–10551.CrossRefGoogle ScholarPubMed
[Soc90] Socolar, J.E.S. (1990). Weak matching rules for quasicrystals, Commun. Math. Phys. 129, 599–619.CrossRefGoogle Scholar
[SS86] Socolar, J.E.S. and Steinhardt, P.J. (1986). Quasicrystals. II. Unit-cell configurations. Phys. Rev. B 34, 617–647.CrossRefGoogle ScholarPubMed
[ST11] Socolar, J.E.S. and Taylor, J.M. (2011). An aperiodic hexagonal tile, J. Comb. Theory A 118, 2207–2231. arXiv:1003.4279.CrossRefGoogle Scholar
[Sol11] Solomon, Y. (2011). Substitution tilings and separated nets with similarities to the integer lattice, Israel J. Math. 181, 445–460. arXiv:0810.5225.CrossRefGoogle Scholar
[Sol97] Solomyak, B. (1997). Dynamics of self-similar tilings, Ergod. Th. & Dynam. Syst. 17, 695–738 and Ergod. Th. & Dynam. Syst. 19 (1999), 1685 (erratum).CrossRefGoogle Scholar
[Sol98a] Solomyak, B. (1998). Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discr. Comput. Geom. 20, 265–279.CrossRefGoogle Scholar
[Sol98b] Solomyak, B. (1998). Spectrum of dynamical systems arising from Delone sets. In [Pat98], pp. 265–275; for an erratum, see the author's homepage.Google Scholar
[Sol07] Solomyak, B. (2007). Pseudo-self-similar tilings in ℝd, J. Math. Sci. (N.Y.) 140, 452–460. arXiv:math/0510014.CrossRefGoogle Scholar
[Sos00] Soshnikov, A. (2000). Determinantal random point fields, Russian Math. Surveys 55, 923–975. arXiv:math.PR/0002099.CrossRefGoogle Scholar
[SO87] Steinhardt, P.J. and Ostlund, S. (eds.) (1987). The Physics of Quasicrystals (World Scientific, Singapore).CrossRef
[Ste17] Steinhaus, H. (1917). Nowa własność mnogości Cantora, Wektor 7, 1–3. English translation: A new property of the Cantor set. In Hugo Steinhaus: Selected Papers, Hartman S. and Urbanik K. (eds.), pp. 205–207 (Polish Academic Publishers, Warsaw, 1985).Google Scholar
[Ste04] Steurer, W. (2004). Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals, Z. Krist. 219, 391–446.Google Scholar
[SD09] Steurer, W. and Deloudi, S. (2009). Crystallograhy of Quasicrystals: Concepts, Methods and Structures (Springer, Berlin).Google Scholar
[SSW07] Steurer, W. and Sutter-Widmer, D. (2007). Photonic and phononic quasicrystals, J. Phys. D: Appl. Phys. 40, R229–R247.CrossRefGoogle Scholar
[Stew89] Stewart, I. (1989). Galois Theory, 2nd ed. (Chapman and Hall, London).CrossRefGoogle Scholar
[ST79] Stewart, I.N. and Tall, D.O. (1979). Algebraic Number Theory (Chapman and Hall, London).CrossRefGoogle Scholar
[STJ89] Strandburg, K.J., Tang, L.-H. and Jarić, M.V. (1989). Phason elasticity in entropic quasicrystals, Phys. Rev. Lett. 63, 314–317.CrossRefGoogle ScholarPubMed
[Str05a] Strungaru, N. (2005). Difiraction Symmetries for Point Sets, PhD thesis (Univ. of Alberta, Edmonton).Google Scholar
[Str05b] Strungaru, N. (2005). Almost periodic measures and long-range order in Meyer sets, Discr. Comput. Geom. 33, 483–505.CrossRefGoogle Scholar
[Str11] Strungaru, N. (2011). Positive definite measures with discrete Fourier transform and pure point Difiraction, Can. Math. Bull. 54 (2011), 544–555.CrossRefGoogle Scholar
[Str13a] Strungaru, N. (2013). Almost periodic measure and Bragg Difiraction, J. Phys. A: Math. Theor. 46, 125205: 1–11. arXiv:1209.2168.CrossRefGoogle Scholar
[Str13b] Strungaru, N. (2013). On the Bragg Difiraction spectra of a Meyer set, Can. J. Math. 65, 675–701. arXiv:1003.3019.CrossRefGoogle Scholar
[SSH02] Suck, J.-B., Schreiber, M. and Häussler, P. (eds.) (2002). Quasicrystals — An Introduction to Structure, Physical Properties and Applications, Springer Series in Materials Science, vol. 55 (Springer, Berlin).
[Tan07] Tan, B. (2007). Mirror substitutions and palindromic sequences, Theor. Comput. Sci. 389, 118–124.CrossRefGoogle Scholar
[Tan90] Tang, L.-H. (1990). Random-tiling quasicrystal in three dimensions, Phys. Rev. Lett. 64, 2390–2393.CrossRefGoogle ScholarPubMed
[Tay10] Taylor, J.M. (2010). Aperiodicity of a functional monotile, available at http://www.math.uni-bielefeld.de/sfb701/preprints/view/420.Google Scholar
[Tem92] Tempelman, A. (1992). Ergodic Theorems for Group Actions (Kluwer, Dordrecht).CrossRefGoogle Scholar
[Ter13] Terauds, V. (2013). The inverse problem of pure point Difiraction — examples and open questions, Preprint arXiv:1303.3260.Google Scholar
[TB13] Terauds, V. and Baake, M. (2013). Some comments on the inverse problem of pure point Difiraction, In Aperiodic Crystals, Schmid, S., Withers, R.L. and Lifshitz, R. (eds.), pp. 35–41 (Springer, Dordrecht). arxiv:1210.3460.Google Scholar
[Tre03] Trebin, H.-R. (ed.) (2003). Quasicrystals: Structure and Physical Properties (Wiley-VCH, Weinheim).CrossRef
[Ush99] Ushakov, N.G. (1999). Selected Topics in Characteristic Functions (Brill Academic Publishers, Utrecht).CrossRefGoogle Scholar
[Vai28] Vaidyanathaswamy, R. (1928). Integer roots of the unit matrix, J. London Math. Soc. 3, 121–124; On the possible periods of integer matrices, J. London Math. Soc. 3, 268–272.Google Scholar
[vDan30] van Dantzig, D. (1930). Ueber topologisch homogene Kontinua, Fund. Math. 15, 102–125.CrossRefGoogle Scholar
[vEdG11] van Enter, A.C.D. and de Groote, E. (2011). An ultrametric state space with a dense discrete overlap distribution: Paperfolding sequences, J. Stat. Phys. 142, 223–228. arXiv:1010.2338.CrossRefGoogle Scholar
[vEHM92] van Enter, A.C.D., Hof, A. and Miȩkisz, J. (1992). Overlap distributions for deterministic systems with many pure states, J. Phys. A: Math. Gen. 25, L1133–L1137.Google Scholar
[vEM92] van Enter, A.C.D. and Miȩkisz, J. (1992). How should one define a (weak) crystal?J. Stat. Phys. 66, 1147–1153.CrossRefGoogle Scholar
[vEMZ98] van Enter, A.C.D., Miȩkisz, J. and Zahradník, M. (1998). Non-periodic longrange order for fast decaying interactions at positive temperatures, J. Stat. Phys. 90, 1441–1447. arXiv:cond-mat/9711206.CrossRefGoogle Scholar
[vEZ97] van Enter, A.C.D. and Zegarlinski, B. (1997). Non-periodic long-range order for one-dimensional pair interactions, J. Phys. A: Math. Gen. 30, 501–505. mp_arc/96-509.Google Scholar
[vSm07] van Smaalen, S. (2007). Incommensurate Crystallography (Oxford University Press, Oxford).CrossRefGoogle Scholar
[vNe32] von Neumann, J. (1933). Zur Operatorenmethode in der klassischen Mechanik, Ann. Math. (2) 33, 587–642.Google Scholar
[Vig80] Vignéras, M.-F. (1980). Arithmétique des Algèbres de Quaternions, LNM 800 (Springer, Berlin).CrossRefGoogle Scholar
[Vin00] Vince, A. (2000). Digit tiling of Euclidean space. In [BM00b], pp. 329–370.Google Scholar
[vQu79] von Querenburg, B. (1979) Mengentheoretische Topologie, 2nd ed. (Springer, Berlin)CrossRefGoogle Scholar
[Wal00] Walters, P. (2000). An Introduction to Ergodic Theory, reprint (Springer, New York).Google Scholar
[WCK87] Wang, M., Chen, H. and Kuo, K.H. (1987). Two-dimensional quasicrystal with eightfold rotational symmetry, Phys. Rev. Lett. 59, 1010–1013.CrossRefGoogle ScholarPubMed
[Wan50] Wannier, G.H. (1950). Antiferromagnetism. The triangular Ising net, Phys. Rev. 79, 357–364 and Phys. Rev. B 7 (1973), 5017 (erratum).CrossRefGoogle Scholar
[Was97] Washington, L.C. (1997). Introduction to Cyclotomic Fields, 2nd ed. (Springer, New York).CrossRefGoogle Scholar
[Wel04] Welberry, T.R. (2004). Diffuse X-Ray Scattering and Models of Disorder (Clarendon Press, Oxford).Google Scholar
[Wey16] Weyl, H. (1916). Über die Gleichverteilung von Zahlen mod. Eins, Math. Annalen 77, 313–352.CrossRefGoogle Scholar
[Wic91] Wicks, K.R. (1991). Fractals and Hyperspaces, LNM 1492 (Springer, Berlin).CrossRefGoogle Scholar
[Wid93] Widom, M. (1993). Bethe ansatz solution of the square-triangle random tiling model, Phys. Rev. Lett. 70, 2094–2097.CrossRefGoogle ScholarPubMed
[Wie50] Wielandt, H. (1950). Unzerlegbare, nicht negative Matrizen, Math. Z. 52, 642–648.CrossRefGoogle Scholar
[Wie27] Wiener, N. (1927). The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions. Part I: The spectrum of an array, J. Math. Massachusetts 6, 145–157.Google Scholar
[Wil74] Williams, R.F. (1974). Expanding attractors, Publ. Math. IHES 43, 169–203.CrossRefGoogle Scholar
[Wit05] Withers, R.L. (2005). Disorder, structured di?use scattering and the transmission electron microscope, Z. Krist. 220, 1027–1034.CrossRefGoogle Scholar
[Wu09] Wu, F.Y. (2009). Exactly Solved Models: A Journey in Statistical Mechanics (World Scientific, Singapore).CrossRefGoogle Scholar
[WMTB76] Wu, T.T., McCoy, B.M., Tracy, C.A. and Barouch, E. (1976). Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Phys. Rev. B 13, 316–374.CrossRefGoogle Scholar
[Zag84] Zagier, D.B. (1984). Zetafunktionen und quadratische Körper (Springer, Berlin).Google Scholar
[Zak02] Zaks, M.A. (2002). On the dimensions of the spectral measure of symmetric binary substitutions, J. Phys. A: Math. Gen. 35, 5833–5841.CrossRefGoogle Scholar
[Zin09] Zint, N. (2009). Influence of Randomness in mathematical models for diffraction and T-cell recognition, PhD thesis (Univ. Bielefeld).Google Scholar
[Zob93a] Zobetz, E. (1993). Homometric cubic point configurations, Z. Krist. 205, 177–199.Google Scholar
[Zob93b] Zobetz, E. (1993). One-dimensional quasilattices: Fractally shaped atomic surfaces and homometry, Acta Cryst. A 49, 667–676.CrossRefGoogle Scholar

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  • References
  • Michael Baake, Universität Bielefeld, Germany, Uwe Grimm, The Open University, Milton Keynes
  • Book: Aperiodic Order
  • Online publication: 18 December 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9781139025256.016
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  • References
  • Michael Baake, Universität Bielefeld, Germany, Uwe Grimm, The Open University, Milton Keynes
  • Book: Aperiodic Order
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  • Chapter DOI: https://doi.org/10.1017/CBO9781139025256.016
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  • Michael Baake, Universität Bielefeld, Germany, Uwe Grimm, The Open University, Milton Keynes
  • Book: Aperiodic Order
  • Online publication: 18 December 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9781139025256.016
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