Some problems on the boundary of fractal geometry and additive combinatorics

Abstract

This paper is an exposition, with some new applications, of our results on the growth of entropy of convolutions. We explain the main result on $\mathbb{R}$, and derive, via a linearization argument, an analogous result for the action of the affine group on $\mathbb{R}$. We also develop versions of the results for entropy dimension and Hausdorff dimension. The method is applied to two problems on the border of fractal geometry and additive combinatorics. First, we consider attractors $X$ of compact families $\Phi$ of similarities of $\mathbb{R}$. We conjecture that if $\Phi$ is uncountable and $X$ is not a singleton (equivalently, $\Phi$ is not contained in a 1-parameter semigroup) then $\dim X=1$. We show that this would follow from the classical overlaps conjecture for self-similar sets, and unconditionally we show that if $X$ is not a point and $\dim\Phi>0$ then $\dim X=1$. Second, we study a problem due to Shmerkin and Keleti, who have asked how small a set $\emptyset\neq Y\subseteq\mathbb{R}$ can be if at every point it contains a scaled copy of the middle-third Cantor set $K$. Such a set must have dimension at least $\dim K$ and we show that its dimension is at least $\dim K+\delta$ for some constant $\delta>0$.