On r-free integers in Beatty sequences

Abstract

Let $r\ge 2$ be a fixed integer. A positive integer n is called r-free if in the canonical representation of n into prime powers each exponent is $<r.$ The integer 1 is considered to be r-free. In this paper, we consider $Q_r(x; \alpha , \beta )$, which is the number of r-free integers of Beatty sequence $\lfloor \alpha n+\beta \rfloor$, $1\le n\le x$, for $\alpha >1$ irrational and with bounded partial quotients, $\beta \in [0,\alpha ).$ We prove that, as $x\rightarrow \infty$

$$\begin{aligned} Q_r(x; \alpha , \beta )={x\over \zeta (r)}+O(x^{(r+1)/2r}\log ^3x ), \end{aligned}$$

which improves Victorovich’s result in the case of square-free integers. Moreover, we also prove there exist infinitely many consecutive square-free numbers of the forms $\lfloor \alpha n+\beta \rfloor$, $\lfloor \alpha n+\beta \rfloor +1$, which improves Dimitrov’s result in 2019.