Dimensions of Furstenberg sets and an extension of Bourgain’s projection theorem

Shmerkin, Pablo; Wang, Hong

doi:10.2140/apde.2025.18.265

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Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem

Pablo Shmerkin and Hong Wang

Vol. 18 (2025), No. 1, 265–278

DOI: 10.2140/apde.2025.18.265

Abstract

We show that the Hausdorff dimension of $(s, t)$ -Furstenberg sets is at least $s + \frac{1}{2} t + 𝜖$ , where $𝜖 > 0$ depends only on $s$ and $t$ . This improves the previously best known bound for $2 s < t \leq 1 + 𝜖 (s, t)$ , in particular providing the first improvement since 1999 to the dimension of classical $s$ -Furstenberg sets for $s < \frac{1}{2}$ . We deduce this from a corresponding discretized incidence bound under minimal nonconcentration assumptions that simultaneously extends Bourgain’s discretized projection and sum-product theorems. The proofs are based on a recent discretized incidence bound of T. Orponen and the first author and a certain duality between $(s, t)$ and $(\frac{1}{2} t, s + \frac{1}{2} t)$ -Furstenberg sets.

Keywords

Furstenberg sets, Bourgain's projection theorem, projections, incidences, Hausdorff dimension, sum-product, discretized sets

Mathematical Subject Classification

Primary: 28A80

Secondary: 28A75, 28A78

Milestones

Received: 11 February 2023

Revised: 23 September 2023

Accepted: 22 October 2023

Published: 15 December 2024

Authors

Pablo Shmerkin
Department of Mathematics University of British Columbia Vancouver, BC Canada

Hong Wang
Courant Institute of Mathematical Sciences New York University New York, NY United States

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Vol. 18, No. 1, 2025