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March 2010 A note on density modulo 1 of certain sets of sums
Roman Urban
Funct. Approx. Comment. Math. 42(1): 29-35 (March 2010). DOI: 10.7169/facm/1269437066


Let $a_1>a_2>1$ and $b_1>b_2>1$ be two distinct pairs of multiplicatively independent integers. If $b_1>a_1$ and $a_2>b_2$ or $b_1<a_1$ and $a_2<b_2$ then we prove that for every $\xi_1,\xi_2,$ with at least one $\xi_i$ irrational, there exists $q\in\mathbb{N}$ such that for any sequence of real numbers $r_m$ the set of sums \[ \{a_1^ma_2^{n}q\xi_1+b_1^mb_2^{n}q\xi_2+r_m:m,n\in\mathbb{N}\}, \] is dense modulo 1. The sets with algebraic numbers $a_i,b_i$ are also considered.


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Roman Urban. "A note on density modulo 1 of certain sets of sums." Funct. Approx. Comment. Math. 42 (1) 29 - 35, March 2010.


Published: March 2010
First available in Project Euclid: 24 March 2010

zbMATH: 1206.11087
MathSciNet: MR2640767
Digital Object Identifier: 10.7169/facm/1269437066

Primary: 11J71
Secondary: 11R04 , 54H20‎

Keywords: Density modulo 1 , multiplicatively independent algebraic numbers. , topological dynamics

Rights: Copyright © 2010 Adam Mickiewicz University

Vol.42 • No. 1 • March 2010
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