Distribution of $\alpha n +\beta $ modulo 1 over integers free from large and small primes
Volume 189 / 2019
Acta Arithmetica 189 (2019), 95-107
MSC: 11K60, 11L07, 11N36.
DOI: 10.4064/aa180218-10-9
Published online: 23 April 2019
Abstract
For any $\varepsilon \gt 0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert \alpha n + \beta \rVert \lt x^{-{1}/{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real numbers $x$. In addition, we also establish an asymptotic formula with an additional square-free condition on $n$. Moreover, if $\alpha$ is quadratic irrational then the asymptotic formulas hold for all sufficiently large $x$.
Our tools come from the Harman sieve which we adapt suitably to sieve for $[y,z]$-smooth numbers. The arithmetic information comes from estimates for exponential sums.
