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

TL;DR

This paper derives asymptotic formulas for the distribution of linear functions modulo 1 over integers with restrictions on prime factors, using Harman's sieve and exponential sum estimates.

Contribution

It introduces new asymptotic formulas for smooth and square-free integers in the context of fractional parts of linear functions, extending previous results.

Findings

Asymptotic formula for smooth integers with fractional parts less than x^(-1/4+ε)

Asymptotic formula with square-free condition on integers

Results hold for all large x when α is quadratic irrational

Abstract

For any $\varepsilon >0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert \alpha n + \beta \rVert < x^{-\frac{1}{4}+\varepsilon} $$ 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.