The uniform Littlewood conjecture fails on a set of positive Hausdorff dimension

Abstract

The uniform Littlewood conjecture (ULC), introduced by Bandi, Fregoli and Kleinbock, asserts in the two-number case that $$ \lim_{Q\to\infty} Q\min_{1\le n\le Q}\|nξ\|\,\|nζ\|=0 $$ for all real $ξ,ζ$. It is proven to hold for almost every pair $(ξ,ζ)$. Schleischitz, however, has recently disproved the full statement and showed that the set of counterexamples contains a dense $G_δ$ set. We prove that a set of counterexample pairs with the first coordinate being a badly approximable number has Hausdorff dimension at least $3/2$. We further show that the set of badly approximable numbers $ξ$ for which there exists $ζ$ such that $(ξ,ζ)$ is a counterexample to ULC has full Hausdorff dimension. This contrasts with the classical Littlewood conjecture, for which the set of possible counterexamples is known to have Hausdorff dimension $0$.