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

Abstract

For any ε > 0, we obtain an asymptotic formula for the number of solutions n ≤ x to αn + β < x −1/4+ε 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 α 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.

Distribution of Integers With Prescribed Structure and Applications

“…The arithmetic information comes from estimates for exponential sums. The results in Chapter 1 have been published in [11].…”