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

Speaker: 

Kam Hung Yau

Affiliation: 

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.

School Seminar Series: