Measure rigidity and p-adic Littlewood-type problems

I will explain how recent work of Lindenstrauss
and Einsiedler-Katok-Lindenstrauss on measure rigidity
of higher rank homogeneous actions gives rise to new
results and conjectures in multiplicative Diophantine
approximation over the product of real and p-adic fields.
Joint with Manfred Einsiedler.