On Furstenberg's intersection conjecture, self-similar measures, and the Lqnorms of convolutions

We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the L-q dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of xp- and xq-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an L-q density for all finite q, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to L-q norms, and likewise relies on an inverse theorem for the decay of L-q norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and it is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemeredi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.