Measure rigidity and $p$-adic Littlewood-type problems
  The paper investigates various $p$-adic versions of Littlewood's conjecture, generalizing a set-up considered recently by de Mathan and Teulie. In many cases it is shown that the sets of exceptions to these conjectures have Hausdorff dimension zero. The proof follows the measure ridigity approach of Einsiedler, Katok and Lindenstrauss. 
Manfred Einsiedler and Dmitry Kleinbock