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An Approximation by Lacunary Sequence of Vectors

Published online by Cambridge University Press:  01 May 2008

ARTŪRAS DUBICKAS
ARTŪRAS DUBICKAS*
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania; Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania (e-mail: arturas.dubickas@mif.vu.lt)
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Abstract

Let be a sequence of real numbers satisfying for each k ≥ 0, where M ≥ 1 is a fixed number. We prove that, for any sequence of real numbers , there is a real number ξ such that for each k ≥ 0. Here, denotes the distance from to the nearest integer. This is a corollary derived from our main theorem, which is a more general matrix version of this statement with explicit constants.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 3 , May 2008 , pp. 339 - 345
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Akhunzhanov, R. K. and Moshchevitin, N. G. (2004) On the chromatic number of a distance graph associated with a lacunary sequence. Dokl. Ross. Akad. Nauk 397 295–296 (in Russian).Google Scholar
[2]Akhunzhanov, R. K. and Moshchevitin, N. G. (2005) Density modulo 1 of sublacunary sequences. Math. Notes 77 741–750.CrossRefGoogle Scholar
[3]Alon, N. and Spencer, J. (1992) The Probabilistic Method, Wiley, New York.Google Scholar
[4]Dubickas, A. (2006) Arithmetical properties of powers of algebraic numbers. Bull. London Math. Soc. 38 70–80.CrossRefGoogle Scholar
[5]Dubickas, A. (2006) On the fractional parts of lacunary sequences. Math. Scand. 99 136–146.CrossRefGoogle Scholar
[6]Dubickas, A. An approximation property of lacunary sequences. Israel J. Math., to appear.Google Scholar
[7]Dubickas, A. On the distribution of powers of a complex number. Submitted.Google Scholar
[8]Erdős, P. (1975) Problems and results on Diophantine approximations II. In Repartition Modulo 1, Actes Colloq. Marseille–Luminy 1974, Vol. 475 of Lecture Notes in Mathematics, Springer, pp. 89–99.Google Scholar
[9]Flatto, L., Lagarias, J. C. and Pollington, A. D. (1995) On the range of fractional parts {ξ(p/q)n}. Acta Arith. 70 125–147.CrossRefGoogle Scholar
[10]Katznelson, Y. (2001) Chromatic numbers of Cayley graphs on ℤ and recurrence. Combinatorica 21 211–219.CrossRefGoogle Scholar
[11]Khintchine, A. (1926) Über eine Klasse linearer diophantischer Approximationen. Rend. Circ. Mat. Palermo 50 170–195.CrossRefGoogle Scholar
[12]deMathan, B. Mathan, B. (1980) Numbers contravening a condition in density modulo 1. Acta Math. Acad. Sci. Hung. 36 237–241.Google Scholar
[13]Moschchevitin, N. G. (2007) A version of the proof for Peres–Schlag's theorem on lacunary sequences. Preprint: arXiv:0708.2087.Google Scholar
[14]Peres, Y. and Schlag, W. (2007) Two Erdős problems on lacunary sequences: Chromatic number and Diophantine approximation. Preprint: arXiv:0706.0223.Google Scholar
[15]Pollington, A. D. (1979) On the density of the sequence {n k ξ}. Illinois J. Math. 23 511–515.CrossRefGoogle Scholar
[16]Weyl, H. (1916) Über die Gleichverteilung von Zahlen modulo Eins. Math. Ann. 77 313–352.CrossRefGoogle Scholar