KLOOSTERMAN SUMS WITH BEATTY SEQUENCES AND APPLICATION

Abstract

Let $\alpha$ and $\beta$ be real numbers. The corresponding Beatty sequence is defined by

$${\mathcal{ℬ}}_{\alpha ,\beta }=\{\lfloor \alpha n+\beta \rfloor :n\in \mathbb{N}\}.$$

We estimate Kloosterman sums with Beatty sequences of the shape

$$\sum _{\begin{array}{c}1\le m\le N\\ (m,q)=1\\ m\in {\mathcal{ℬ}}_{\alpha ,\beta }\end{array}}{e}_q(rm+s\overline{m}),$$

where $q$ is an integer. When $q=p$ is a prime, bounds for exponential sums with rational functions along Beatty sequences are also obtained. Furthermore, bounds for $\text{max}\{m,\tilde{m}\}$ subject to $m,\tilde{m}\in \mathbb{Z}\cap [1,p)$, $c$ indivisible by $p$, $m\tilde{m}\equiv c(\text{mod\hspace{0.17em}}p)$ and $m$ belonging to some fixed Beatty sequence are improved.