## Distribution of $\alpha n +\beta $ modulo 1 over integers free from large and small primes

### Volume 189 / 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.