Furstenberg sets estimate in the plane

TL;DR

This paper proves the Furstenberg set conjecture in the plane, establishing precise lower bounds for Hausdorff dimension, and applies these results to sum-product problems and projection questions.

Contribution

It fully resolves the Furstenberg set conjecture in , providing exact dimension bounds and connecting to sum-product estimates and projection problems.

Findings

Furstenberg sets in  have Hausdorff dimension at least min(s+t, (3s+t)/2, s+1)

Derived an Elekes-type bound for discretized sum-product problem

Resolved an orthogonal projection question of Oberlin

Abstract

We fully resolve the Furstenberg set conjecture in $\mathbb{R}^2$, that a $(s, t)$-Furstenberg set has Hausdorff dimension $\ge \min(s+t, \frac{3s+t}{2}, s+1)$. As a result, we obtain an analogue of Elekes' bound for the discretized sum-product problem and resolve an orthogonal projection question of Oberlin.