On an effective version of Furstenberg’s theorem

(joint with J. Bourgain, Ph. Michel, A. Venkatesh)

In a paper that influenced many aspects of modern ergodic theory and
dynamical systems, Furstenberg proved in particular that any subset of
${\bf R}/{\bf Z}$ invariant under multiplication by two multiplicatively independent
integer is either finite or dense. This turned out to be the starting
point of a very fruitful direction of research into rigidity properties
of multiparameter diagonalizable actions. I will describe a quantitative
version of this theorem that rests on a quantitative version of a
related measure theoretic result by D. Rudolph.