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

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 addi