A note on density modulo 1 of certain sets of sums (Q964225)

Let \(a_1>a_2>1\) and \(b_1>b_2>1\) be two pairs of multiplicatively independent integers. Suppose that \(a_1<b_1\) and \(a_2>b_2\). The author proves that for any real numbers \(\xi_1,\xi_2\) with at least one of them irrational, there exists \(q\in{\mathbb N}\) such that for any sequence of real numbers \(r_m\), the set \[ \{a_1^ma_2^nq\xi_1+b_1^mb_2^nq\xi_2+r_m:m,n\in{\mathbb N}\} \] is dense modulo \(1\). The result is then formulated and proved for algebraic numbers \(a_i\) and \(b_i\).