Measure rigidity and p-adic Littlewood-type problems

Various -adic versions of Littlewood's conjecture are investigated, generalizing a set-up considered recently by de Mathan and Teulié. 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.