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

Abstract

For any $\epsilon >0$, we obtain an asymptotic formula for the number of solutions $n\le x$ to $$\Vert \alpha n+\beta \Vert <x^{-\frac{1}{4}+\epsilon }$$ where $n$ is $[y,z]$-smooth for infinitely many real number $x$. 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 holds for all sufficiently large $x$. Our ingredients 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.

arXiv commentFixed minor typos