Student Number Theory Seminar

Jérémy Champagne, University of Waterloo

Small fractional parts of polynomials (aka 11J54)

In the eary 1900's, Hardy and Littlewood asked the following question: given a real number α and integer k>1, what is the smallest distance obtained between αn^k and the nearest integer as n runs over the set {1,...,N}? More specifically, does there exist an exponent theta_k>0 such that the smallest distance is at most N^-theta_k for sufficiently large N? This question was answered positively by Vinogradov a couple decades later, but the question of finding the largest possible theta_k with this property is still open.

In this talk, I will discuss some historical results around this problem and present some typical methods used in the literature.

MC 5479