A randomized version of the Littlewood Conjecture

TL;DR

This paper introduces a randomized approach to the Littlewood Conjecture, proving that a modified version holds for almost all cases, with an even stronger result involving a logarithmic factor.

Contribution

It presents a novel randomized covering method that confirms a stronger form of the Littlewood Conjecture for almost all center translations.

Findings

The randomized version of the Littlewood Conjecture is true for almost all centers.

An enhanced statement with an additional logarithmic factor also holds.

The approach uses a probabilistic covering argument.

Abstract

The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each hyperbola by a random amount which depends on the denominator of the corresponding rational. Using a randomized covering argument we prove that, not only is this randomized version of the Littlewood Conjecture true for almost all choices of centers, an even stronger statement with an extra factor of a logarithm also holds.