On self-similar sets with overlaps and inverse theorems for entropy

We study the dimension of self-similar sets and measures on the line. We show that if the dimension is less than the generic bound of $\mathrm{min}\{1,s\}$, where $s$ is the similarity dimension, then there are superexponentially close cylinders at all small enough scales. This is a step towards the conjecture that such a dimension drop implies exact overlaps and confirms it when the generating similarities have algebraic coefficients. As applications we prove Furstenberg’s conjecture on projections of the one-dimensional Sierpinski gasket and achieve some progress on the Bernoulli convolutions problem and, more generally, on problems about parametric families of self-similar measures. The key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale $2^{-n}$ has only a small amount of growth under convolution.