On $k$-free numbers over Beatty sequences

Abstract

In this paper, we consider k-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number α > 1 of finite type τ < ∞, any constant ε > 0, we can show thatwhere Q k is the set of positive k-free integers and the implied constant depends only on α, ε, k and β. This improves previous results. The main new ingredient of our idea is an employing of the double exponential sums of the type 1≤h≤H 1≤n≤x n∈Q k e(ϑhn).