On the values of Beatty sequence in an arithmetic progression

Abstract

In the paper, we consider $N_d(x)=N(x;\alpha,\beta;d,a)$, $x\in\mathbb{N}$, which is the number of values of Beatty sequence $[\alpha n+\beta]$, $1\leqslant n\leqslant x$, for $\alpha>1$ irrational and with bounded partial quotients, $\beta\in[0;\alpha)$, in an arithmetic progression $(a+kd)$\textup, $k\in\mathbb{N}$. We prove the asymptotic formula $N_d(x) = \frac{x}{d} + O(d\ln^3 x)$ as $x\to\infty,$
where the implied constant is absolute. For growing difference~$d$ the~result is non-trivial provided $d\ll \sqrt{x}\ln^{-3/2-\varepsilon}x$, $\varepsilon>0$.