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We study a class of measures on the real line with a kind of self-similarstructure, which we call dynamically driven self-similar measures, and containproper self-similar measures such as Bernoulli convolutions as special cases.Our main result gives an expression for the $L^q$-dimensions of suchdynamically driven self-similar measures, under certain conditions. As anapplication, we settle Furstenberg's long-standing conjecture on the dimensionof the intersections of $\times p$ and $\times q$-invariant sets. Among severalother 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 thedimensions of self-similar measures and his inverse theorem for entropy. Ourmethod 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 $L^q$ normsof discrete measures under convolution. This central piece of our approach maybe of independent interest, and is an application of well-known methods andresults in additive combinatorics: the asymmetric version of theBalog-Szemer\'{e}di-Gowers Theorem due to Tao-Vu, and some constructions ofBourgain.

On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions