Abstract
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 < ∞ and any constant ε > 0, we can show that
where Qk 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 employing double exponential sums of the type
.
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I am deeply grateful to the referee(s) for carefully reading the manuscript and making useful suggestions.
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Zhang, W. On k-free numbers over Beatty sequences. Czech Math J 73, 839–847 (2023). https://doi.org/10.21136/CMJ.2023.0304-22
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DOI: https://doi.org/10.21136/CMJ.2023.0304-22