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

Abstract

We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at small scales there are super-exponentially close cylinders. This is a step towards the folklore conjecture that such a drop in dimension is explained only by exact overlaps, and confirms the conjecture in cases where the contraction parameters are algebraic. It also gives an affirmative answer to a conjecture of Furstenberg, showing that the projections of the "1-dimensional Sierpinski gasket" in irrational directions are all of dimension 1. As another consequence, if a family of self-similar sets or measures is parametrized in a real-analytic manner, then, under an extremely mild non-degeneracy condition, the set of "exceptional" parameters has Hausdorff dimension 0. Thus, for example, there is at most a zero-dimensional set of parameters 1/2<r<1 such that the corresponding Bernoulli convolution has dimension <1, and similarly for Sinai's problem on iterated function systems that contract on average. A central ingredient of the proof is an inverse theorem for the growth of Shannon entropy of convolutions of probability measures. For the dyadic partition D_n of the line into intervals of length 1/2^n, we show that if H(nu*mu,D_n)/n < H(mu,D_n)/n + delta for small delta and large n, then, when restricted to random element of a partition D_i, 0<i<n, either mu is close to uniform or nu is close to atomic. This should be compared to results in additive combinatorics that give the global structure of measures satisfying H(nu*mu,D_n)/n < H(mu,D_n)/n + O(1/n).