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

Kam Hung Yau

UNSW

## Date:

Wed, 13/09/2017 - 3:00pm

## Venue:

RC-4082, The Red Centre, UNSW

## Abstract:

For any sufficiently small real number $\varepsilon >0$, we obtain an asymptotic formula for the number of solutions to $\lVert \alpha n + \beta \rVert < x^{\varepsilon}$ ($\lVert \cdot \rVert$ is the nearest integer function) where $n \le x$ is square-free with prime factors in $[y,z] \subseteq [1,x]$ for infinitely many real number $x$. If $\alpha$ is eventually periodic then it holds for all positive real numbers $x$.

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.