Title：Positive Logarithmic Hausdorff Measures of Exceptional Sets for the p-adic and t-adic Littlewood Conjectures

Speaker：Dzmitry Badziahin (Professor，The University of Sydney)

Time:  Sep 28,  2026, 10:00-11:00

Venue: Room 3A02, Building No. 37, Wushan Campus

Abstract: 

In 2004 de Mathan and Teulie proposed several analogues of the famous Littlewood conjecture in a hope that they may be easier to settle and that they will provide some insight into the proof of the Littlewood conjecture itself. The first one is the so called p-adic Littlewood conjecture (PLC). It says that for all primes p and all real x, one always has

$$\liminf_{q\to\infty} q\cdot |q|_p\cdot ||qx|| = 0,$$

where $|q|_p$ is the p-adic norm of q and ||qx|| is the distance from qx to the nearest integer. This conjecture is still open, however, due to work of Einsiedler and Kleinbock, it is know that the set of its counterexamples x has zero Hausdorff dimension, i.e. is very small. I will talk about what is known about the conjecture and the set of its putative counterexamples. In particular, if this set is nonempty it can not be too small, i.e. certain its logarithmic Hausdorff measures must be positive.

Next, we will also consider the so-called t-adic Littlewood conjecture (TLC) which is a function field analogue of the PLC. Due to works of Adicean, Lai, Lunnon, Nesharim, Robertson and the speaker, this conjecture fails for all finite fields of odd characteristics. We will show that the sets of its counterexamples have positive logarithmic Hausdorff measures.