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December 2008 Equidistribution modulo 1 and Salem numbers
Christophe Doche, Michel Mendès France, Jean-Jacques Ruch
Funct. Approx. Comment. Math. 39(2): 261-271 (December 2008). DOI: 10.7169/facm/1229696575


Let $\theta$ be a Salem number. It is well-known that the sequence $(\theta^n)$ modulo 1 is dense but not equidistributed. In this article we discuss equidistributed subsequences. Our first approach is computational and consists in estimating the supremum of $\lim_{n\to\infty} n/s(n)$ over all equidistributed subsequences $(\theta^{s(n)})$. As a result, we obtain an explicit upper bound on the density of any equidistributed subsequence. Our second approach is probabilistic. Defining a measure on the family of increasing integer sequences, we show that relatively to that measure, almost no subsequence is equiditributed.


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Christophe Doche. Michel Mendès France. Jean-Jacques Ruch. "Equidistribution modulo 1 and Salem numbers." Funct. Approx. Comment. Math. 39 (2) 261 - 271, December 2008.


Published: December 2008
First available in Project Euclid: 19 December 2008

zbMATH: 1211.11091
MathSciNet: MR2490740
Digital Object Identifier: 10.7169/facm/1229696575

Primary: 11K06
Secondary: 11J71

Keywords: $J_0$ Bessel function , Equidistribution modulo 1 , Salem number

Rights: Copyright © 2008 Adam Mickiewicz University

Vol.39 • No. 2 • December 2008
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