On the distribution of \alpha n + \beta modulo 1

Abstract: 

For any sufficiently small real number ε>0ε>0, we obtain an asymptotic formula for the number of solutions to ∥αn+β∥<xε‖αn+β‖<xε (∥⋅∥‖⋅‖ is the nearest integer function) where n≤xn≤x is square-free with prime factors in [y,z]⊆[1,x][y,z]⊆[1,x] for infinitely many real number xx. If αα is eventually periodic then it holds for all positive real numbers xx. 

The method we use is the Harman sieve where the arithmetic information comes from bounds of exponential sums over square-free integers. I will talk about the the history and heuristics on the problem and provide a sketch of the theorem and the main lemmas associated with it.