A combinatorial proof of a sumset conjecture of Furstenberg

TL;DR

This paper presents a new combinatorial proof of Furstenberg's sumset conjecture, establishing the Hausdorff dimension of sumsets of invariant sets under different scales without using entropy methods.

Contribution

The authors provide a combinatorial proof of Furstenberg's sumset conjecture, avoiding entropy techniques and introducing a discrete Marstrand projection theorem and subtree regularity theorem.

Findings

Proved that the Hausdorff dimension of the sumset equals the minimum of 1 and the sum of individual dimensions.

Established uniform bounds on the size of sumsets across parameters at fixed scales.

Developed combinatorial tools that may be of independent interest for fractal geometry.

Abstract

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $\log r / \log s$ is irrational and $X$ and $Y$ are $\times r$- and $\times s$-invariant subsets of $[0,1]$, respectively, then $\dim_\text{H} (X+Y) = \min ( 1, \dim_\text{H} X + \dim_\text{H} Y)$. Our main result yields information on the size of the sumset $\lambda X + \eta Y$ uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.