Functiones et Approximatio Commentarii Mathematici

A note on density modulo 1 of certain sets of sums

Roman Urban

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Abstract

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.

Article information

Source
Funct. Approx. Comment. Math., Volume 42, Number 1 (2010), 29-35.

Dates
First available in Project Euclid: 24 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.facm/1269437066

Digital Object Identifier
doi:10.7169/facm/1269437066

Mathematical Reviews number (MathSciNet)
MR2640767

Zentralblatt MATH identifier
1206.11087

Subjects
Primary: 11J71: Distribution modulo one [See also 11K06]
Secondary: 11R04: Algebraic numbers; rings of algebraic integers 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

Keywords
Density modulo 1 topological dynamics multiplicatively independent algebraic numbers.

Citation

Urban, Roman. A note on density modulo 1 of certain sets of sums. Funct. Approx. Comment. Math. 42 (2010), no. 1, 29--35. doi:10.7169/facm/1269437066. https://projecteuclid.org/euclid.facm/1269437066


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