Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T02:29:50.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Gaps problems and frequencies of patches in cut and project sets

Published online by Cambridge University Press:  03 March 2016

ALAN HAYNES ,
HENNA KOIVUSALO ,
JAMES WALTON  and
LORENZO SADUN
ALAN HAYNES
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: [email protected]; [email protected]; [email protected]
HENNA KOIVUSALO
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: [email protected]; [email protected]; [email protected]
JAMES WALTON
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: [email protected]; [email protected]; [email protected]
LORENZO SADUN
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX 78712-1082, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 1 , July 2016 , pp. 65 - 85
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Beresnevich, V., Dickinson, D. and Velani, S.Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc. 179 (2006), no. 846.Google Scholar
[2]Berthé, V. and Vuillon, L.. Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences. Discrete Math. 223 (2000), no. 1–3, 27–53.Google Scholar
[3]Bleher, P. M., Homma, Y., Ji, L. L., Roeder, R. K. W. and Shen, J. D.. Nearest neighbor distances on a circle: multidimensional case. J. Stat. Phys. 146 (2012), no. 2, 446–465.Google Scholar
[4]Broderick, R., Fishman, L., Kleinbock, D., Reich, A. and Weiss, B.. The set of badly approximable vectors is strongly C 1 incompressible. Math. Proc. Camb. Phil. Soc. 153 (2012), no. 2, 319–339.Google Scholar
[5]Broderick, R., Fishmann, L. and Simmons, D.Badly approximable systems of affine forms and incompressibility on fractals. J. Number Theory 133 (2013), no. 7, 2186–2205.CrossRefGoogle Scholar
[6]Cassels, J. W. S.An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics, no. 45 (Cambridge University Press, New York, 1957).Google Scholar
[7]Cobeli, C., Groza, G., Vâjâitu, M. and Zaharescu, A.. Generalization of a theorem of Steinhaus. Colloq. Math. 92 (2002), no. 2, 257–266.Google Scholar
[8]de Bruijn, N. G.Algebraic theory of Penrose's nonperiodic tilings of the plane I, II. Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 1, 39–66.Google Scholar
[9]Falconer, K.Fractal Geometry: Mathematical Foundations and Applications (Wiley, New York, 2003).CrossRefGoogle Scholar
[10]Fraenkel, A. S. and Holzman, R.Gaps problems for integer part and fractional part sequences. J. Number Theory 50 (1995), 66–86.Google Scholar
[11]Geelen, J. F. and Simpson, R. J.A two-dimensional Steinhaus theorem. Australas. J. Combin. 8 (1993), 169–197.Google Scholar
[12]Haynes, A., Kelly, M. and Weiss, B. Equivalence relations on separated nets arising from linear toral flows. Proc. Lond. Math. Soc. (2014), to appear http://arxiv.org/abs/1211.2606.Google Scholar
[13]Julien, A.Complexity and cohomology for cut-and-projection tilings. Ergodic Theory Dyn. Syst. 30 (2010), no. 2, 489–523.Google Scholar
[14]McMullen, C. T.Winning sets, quasiconformal maps and Diophantine approximation. Geom. Funct. Anal. 20 (2010), no. 3, 726–740.Google Scholar
[15]Moody, R.Model sets: A survey, in From Quasicrystals to More Complex Systems, eds. Axel, F., Dénoyer, F., Gazeau, J. P.. Centre de physique Les Houches (Springer-Verlag, 2000).Google Scholar
[16]Schmidt, W. M.On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 178–199.CrossRefGoogle Scholar
[17]Schmidt, W. M.Badly approximable systems of linear forms. J. Number Theory 1 (1969), 139–154.CrossRefGoogle Scholar
[18]Senechal, M.Quasicrystals and geometry (Cambridge University Press, Cambridge, 1995).Google Scholar
[19]Shechtman, D., Blech, I., Gratias, D. and Cahn, J. W.. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 52 (1984), no.20, 1951–1954.Google Scholar
[20]Slater, N. B.The distribution of the integers N for which {θN} < φ. Proc. Camb. Phil. Soc. 46 (1950), 525–534.CrossRefGoogle Scholar
[21]Slater, N. B.Theory of Unimoloecular Reactions (Cornell University Press, New York, 1959).Google Scholar
[22]Sós, V. T.. On the theory of diophantine approximations I. Acta Math. Acad. Sci. Hungar. 8 (1957), 461–472.Google Scholar
[23]Świerczkowski, S.. On successive settings of an arc on the circumference of a circle. Fund. Math. 46 (1958), 187–189.CrossRefGoogle Scholar