Smooth discrepancy and Littlewood’s conjecture

Abstract: Given \boldsymbol \alpha \in [0,1]^d, we estimate the smooth discrepancy of the Kronecker sequence (n \boldsymbol \alpha \: \mathrm{mod} \: 1)_{n=1}^\infty. We find that it can be smaller than the classical discrepancy of any sequence when d \le 2, and can even be bounded in the case d=1. To achieve this, we establish a novel deterministic analogue of Beck’s local-to-global principle (Annals 1994), which relates the discrepancy of a Kronecker sequence to multiplicative diophantine approximation. This opens up a new avenue of attack for Littlewood’s conjecture.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar