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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

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Let $(t_k)_{k=0}^{\infty}$ be a sequence of real numbers satisfying $t_0 \ne 0$ and $|t_{k+1}| \geq (1+1/M) |t_{k}|$ for each k ≥ 0, where M ≥ 1 is a fixed number. We prove that, for any sequence of real numbers $(\xi_k)_{k=0}^{\infty}$, there is a real number ξ such that $\|t_k \xi-\xi_k\|>1/(80M \log(28M))$ for each k ≥ 0. Here, $\|x\|$ denotes the distance from $x \in \R$ 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.

Information

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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