  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 $\times p$ and $\times q$-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 is an application of well-known methods and
results in additive combinatorics: the asymmetric version of the
Balog-Szemer\'{e}di-Gowers Theorem due to Tao-Vu, and some constructions of
Bourgain.
