Smooth numbers in Beatty sequences
Tom 200 / 2021
Acta Arithmetica 200 (2021), 429-438
MSC: Primary 11N25; Secondary 11L03.
DOI: 10.4064/aa210322-22-6
Opublikowany online: 6 October 2021
Streszczenie
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.
