Smooth discrepancy and Littlewood's conjecture

TL;DR

This paper introduces a new method to estimate the smooth discrepancy of Kronecker sequences, revealing potential for improved bounds in low dimensions and offering a novel approach to Littlewood's conjecture through a deterministic analogue of Beck's principle.

Contribution

It establishes a deterministic analogue of Beck's local-to-global principle relating discrepancy to Diophantine approximation, enabling new insights into Littlewood's conjecture.

Findings

Smooth discrepancy can be smaller than classical discrepancy in dimensions up to 2.

In dimension 1, the smooth discrepancy can be bounded.

A new deterministic analogue of Beck's local-to-global principle is developed.

Abstract

Given $\boldsymbol{\alpha} \in [0,1]^d$, we estimate the smooth discrepancy of the Kronecker sequence $(n \boldsymbol{\alpha} \,\mathrm{mod}\, 1)_{n\geq 1}$. We find that it can be smaller than the classical discrepancy of $\textbf{any}$ 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 (Ann. of Math. 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.