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Furstenberg sets estimate in the plane

Kevin Ren, Hong Wang

TL;DR

This work resolves the planar Furstenberg conjecture by proving that any (s,t)-Furstenberg set E in R^2 has Hausdorff dimension at least $\dim_H E \ge \min\{s+t, \frac{3s+t}{2}, s+1\}$ for s∈(0,1], t∈(0,2]. The authors develop a robust multi-scale incidence framework using a high-low decomposition of tube incidences, refine configurations across scales, and prove a discretized Furstenberg theorem that yields sharp bounds in the δ-discretized setting. They then treat the general case by combining semi-well-spaced and almost-AD-regular regimes via Orponen-Shmerkin’s results, culminating in a complete resolution. Consequences include a discretized sum-product bound in the plane and a resolution of Oberlin’s exceptional projection conjecture, linking Furstenberg geometry with discretized additive combinatorics and projection theory. The approach advances the strategy of induction on scales and multi-scale decompositions in geometric measure theory and incidence geometry, with potential applications to related problems in fractal geometry and harmonic analysis.

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.

Furstenberg sets estimate in the plane

TL;DR

This work resolves the planar Furstenberg conjecture by proving that any (s,t)-Furstenberg set E in R^2 has Hausdorff dimension at least for s∈(0,1], t∈(0,2]. The authors develop a robust multi-scale incidence framework using a high-low decomposition of tube incidences, refine configurations across scales, and prove a discretized Furstenberg theorem that yields sharp bounds in the δ-discretized setting. They then treat the general case by combining semi-well-spaced and almost-AD-regular regimes via Orponen-Shmerkin’s results, culminating in a complete resolution. Consequences include a discretized sum-product bound in the plane and a resolution of Oberlin’s exceptional projection conjecture, linking Furstenberg geometry with discretized additive combinatorics and projection theory. The approach advances the strategy of induction on scales and multi-scale decompositions in geometric measure theory and incidence geometry, with potential applications to related problems in fractal geometry and harmonic analysis.

Abstract

We fully resolve the Furstenberg set conjecture in , that a -Furstenberg set has Hausdorff dimension . As a result, we obtain an analogue of Elekes' bound for the discretized sum-product problem and resolve an orthogonal projection question of Oberlin.
Paper Structure (14 sections, 19 theorems, 115 equations)

This paper contains 14 sections, 19 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Exceptional set estimate for orthogonal projections
  3. Discretized sum-product
  4. Roadmap
  5. Structure of the paper
  6. Acknowledgments
  7. preliminaries
  8. Thickening and high-low method
  9. Pigeonholing
  10. Main Theorem on Furstenberg sets
  11. The case $s + t \geq 2$
  12. When $\mathcal{P}$ is $(2-s)$-dimensional at high scales and $s$-dimensional at low scales
  13. Proof of Lemma \ref{['lem:r-rich']}
  14. General case using Orponen-Shmerkin's regular Furstenberg result

Key Result

Theorem 1.1

Fix $s \in (0, 1]$ and $t \in (0, 2]$. Any $(s, t)$-Furstenberg set $F \subset \mathbb{R}^2$ has Hausdorff dimension

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3: $(\delta, s, C)$-set
  • Definition 1.4: AD-regular set
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1: Dyadic cubes
  • Definition 2.2
  • Definition 2.3: Dyadic $\delta$-tubes
  • Definition 2.4: Slope set
  • ...and 43 more