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 $$\underset{Q\to \infty }{\lim }Q\underset{1\le n\le Q}{\min }\Vert n\xi \Vert \Vert n\zeta \Vert =0$$ for all real $\xi ,\zeta$. It is proven to hold for almost every pair $(\xi ,\zeta )$. Schleischitz, however, has recently disproved the full statement and showed that the set of counterexamples contains a dense $G_{\delta }$ 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 $\xi$ for which there exists $\zeta$ such that $(\xi ,\zeta )$ 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 $0$.

19 pages, comments welcome