Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

Let $\theta$ be an irrational number of finite type and let $\psi \ge 0$. We consider numbers in the Beatty sequence of integer parts, $$\mathcal B(x) = \{\lfloor \theta n + \psi \rfloor : 1 \le n \le x\}.$$ Let $C \gt 3$. Writing $P(n)$ for the largest prime factor of $n$ and $|\ldots |$ for cardinality, we show that $$|\{n\in \mathcal B(x) : P(n) \le y\}| = \frac 1\theta \, \Psi (\theta x, y) (1 + o(1))$$ as $x\to \infty$, uniformly for $y \ge (\log x)^C$. Here $\Psi (X,y)$ denotes the number of integers up to $X$ with $P(n) \le y$. The range of $y$ extends that given by Akbal (2020). The work of Harper (2016) plays a key role in the proof.