On multiplicatively badly approximable numbers

TL;DR

This paper demonstrates that the Littlewood Conjecture fails when multiplied by log q and loglog q, showing that the set of pairs with a positive liminf in this context has full dimension.

Contribution

It introduces a new perspective on multiplicatively badly approximable numbers by showing the conjecture's failure under logarithmic factors.

Findings

The set of (x,y) with positive liminf has full Hausdorff dimension.

The Littlewood Conjecture does not hold when scaled by log q and loglog q.

Provides insight into the structure of multiplicatively badly approximable pairs.

Abstract

The Littlewood Conjecture states that liminf_{q\to \infty} q . ||qx|| . ||qy|| = 0 for all pairs (x,y) of real numbers. We show that with the additional factor of log q . loglog q the statement is false. Indeed, our main result implies that the set of (x,y) for which liminf_{q\to\infty} q . log q . loglog q . ||qx|| . ||qy|| > 0 is of full dimension.