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

TL;DR

This paper investigates the Hausdorff dimension of self-similar sets with overlaps, proving that dimension drops are due to exact overlaps in algebraic cases, and establishes an inverse entropy theorem with broad implications for fractal geometry.

Contribution

It introduces an inverse theorem for entropy growth of convolutions, linking dimension drops to exact overlaps and confirming conjectures for algebraic parameters.

Findings

Dimension drop linked to super-exponentially close cylinders

Proves projections of the 1D Sierpinski gasket have full dimension in irrational directions

Shows the set of exceptional parameters for Bernoulli convolutions has Hausdorff dimension zero

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).