A note on density modulo 1 of certain sets of sums

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.