Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-27T20:49:18.825Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE BINARY DIGITS OF ALGEBRAIC NUMBERS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  13 October 2010

HAJIME KANEKO
HAJIME KANEKO*
Affiliation:
Department of Mathematics, Kyoto University, Oiwake-tyou, Kitashirakawa, Kyoto-shi, Kyoto 606-8502, Japan (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Borel conjectured that all algebraic irrational numbers are normal in base 2. However, very little is known about this problem. We improve the lower bounds for the number of digit changes in the binary expansions of algebraic irrational numbers.

Keywords

algebraic numbersbinary expansionsnonadjacent form

MSC classification

Secondary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11K60: Diophantine approximation
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 2 , October 2010 , pp. 233 - 244
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Bailey, D. H., Borwein, J. M., Crandall, R. E. and Pomerance, C., ‘On the binary expansions of algebraic numbers’, J. Théor. Nombres Bordeaux 16 (2004), 487–518.Google Scholar
[2]Booth, A. D., ‘A signed multiplication technique’, Quart. J. Mech. Appl. Math. 4 (1951), 236–240.CrossRefGoogle Scholar
[3]Borel, É., ‘Les probabilités dénombrables et leurs applications arithmétiques’, Rend. Circ. Mat. Palermo 27 (1909), 247–271.CrossRefGoogle Scholar
[4]Borel, É., ‘Sur les chiffres décimaux de et divers problèmes de probabilités en chaîne’, C. R. Acad. Sci. Paris 230 (1950), 591–593.Google Scholar
[5]Bosma, W., ‘Signed bits and fast exponentiation’, J. Théor. Nombres Bordeaux 13 (2001), 27–41.CrossRefGoogle Scholar
[6]Bugeaud, Y., ‘On the b-ary expansion of an algebraic number’, Rend. Sem. Mat. Univ. Padova 118 (2007), 217–233.Google Scholar
[7]Bugeaud, Y. and Evertse, J.-H., ‘On two notions of complexity of algebraic numbers’, Acta Arith. 133 (2008), 221–250.Google Scholar
[8]Dajani, K., Kraaikamp, C. and Liardet, P., ‘Ergodic properties of signed binary expansions’, Discrete Contin. Dyn. Syst. 15 (2006), 87–119.Google Scholar
[9]Evertse, J.-H. and Schlickewei, H. P., ‘A quantitative version of the absolute subspace theorem’, J. Reine Angew. Math. 548 (2002), 21–127.Google Scholar
[10]Grabner, P. J. and Heuberger, C., ‘On the number of optimal base 2 representations of integers’, Des. Codes Cryptogr. 40 (2006), 25–39.Google Scholar
[11]Heuberger, C., ‘Minimal expansions in redundant number systems: Fibonacci bases and greedy algorithms’, Period. Math. Hungar. 49 (2004), 65–89.CrossRefGoogle Scholar
[12]Locher, H., ‘On the number of good approximations of algebraic numbers by algebraic numbers of bounded degree’, Acta Arith. 89 (1999), 97–122.Google Scholar
[13]Massey, J. L. and Garcia, O. N., ‘Error-correcting codes in computer arithmetic’, in: Advances in Information Systems Science, Vol. 4, (ed. Tou, J.) (Plenum Press, New York, 1972), pp. 273–326.CrossRefGoogle Scholar
[14]Morain, F. and Olivos, J., ‘Speeding up the computations on an elliptic curve using addition–subtraction chains’, RAIRO Inform. Théor. Appl. 24 (1990), 531–543.Google Scholar
[15]Reitwiesner, G. W., ‘Binary arithmetic’, Adv. Comput. 1 (1960), 231–308.Google Scholar
[16]Ridout, D., ‘Rational approximations to algebraic numbers’, Mathematika 4 (1957), 125–131.Google Scholar
[17]Rivoal, T., ‘On the bits counting function of real numbers’, J. Aust. Math. Soc. 85 (2008), 95–111.CrossRefGoogle Scholar