Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane

Tuomas Orponen, Pablo Shmerkin

TL;DR

The paper advances the Furstenberg set problem and projection theory in the plane by proving ε(s,t)>0 improvements for all s∈(0,1) and t∈(s,2], showing that (s,t)-Furstenberg sets have dim_H(K)≥2s+ε and that for analytic K of dim_H(K)=t, the set of directions with small projections has dim_H≤s−ε. The authors reduce these to a single δ-discretised incidence statement, and develop a novel induction-on-scales framework, a multiscale decomposition for general δ-sets, and a covering-lemma that preserves separations across scales. Their method combines point-line duality, (δ,s)-set structure, and incidence counting to achieve ε-improvements beyond Wolff-type bounds, with implications for related problems such as Kaufman-type projection theorems and connections to discretized sum-product phenomena. The results unify Furstenberg-type and projection-type problems under a common discretised paradigm and are robust to compact-subset uniformity, paving the way for further quantitative gains in planar fractal geometry.

Abstract

Let $0 \leq s \leq 1$ and $0 \leq t \leq 2$. An $(s,t)$-Furstenberg set is a set $K \subset \mathbb{R}^{2}$ with the following property: there exists a line set $\mathcal{L}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{L} \geq t$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\ell \in \mathcal{L}$. We prove that for $s\in (0,1)$, and $t \in (s,2]$, the Hausdorff dimension of $(s,t)$-Furstenberg sets in $\mathbb{R}^{2}$ is no smaller than $2s + ε$, where $ε> 0$ depends only on $s$ and $t$. For $s>1/2$ and $t = 1$, this is an $ε$-improvement over a result of Wolff from 1999. The same method also yields an $ε$-improvement to Kaufman's projection theorem from 1968. We show that if $s \in (0,1)$, $t \in (s,2]$ and $K \subset \mathbb{R}^{2}$ is an analytic set with $\dim_{\mathrm{H}} K = t$, then $$\dim_{\mathrm{H}} \{e \in S^{1} : \dim_{\mathrm{H}} π_{e}(K) \leq s\} \leq s - ε,$$ where $ε> 0$ only depends on $s$ and $t$. Here $π_{e}$ is the orthogonal projection to $\mathrm{span}(e)$.

On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane

TL;DR

The paper advances the Furstenberg set problem and projection theory in the plane by proving ε(s,t)>0 improvements for all s∈(0,1) and t∈(s,2], showing that (s,t)-Furstenberg sets have dim_H(K)≥2s+ε and that for analytic K of dim_H(K)=t, the set of directions with small projections has dim_H≤s−ε. The authors reduce these to a single δ-discretised incidence statement, and develop a novel induction-on-scales framework, a multiscale decomposition for general δ-sets, and a covering-lemma that preserves separations across scales. Their method combines point-line duality, (δ,s)-set structure, and incidence counting to achieve ε-improvements beyond Wolff-type bounds, with implications for related problems such as Kaufman-type projection theorems and connections to discretized sum-product phenomena. The results unify Furstenberg-type and projection-type problems under a common discretised paradigm and are robust to compact-subset uniformity, paving the way for further quantitative gains in planar fractal geometry.

Abstract

Let and . An -Furstenberg set is a set with the following property: there exists a line set of Hausdorff dimension such that for all . We prove that for , and , the Hausdorff dimension of -Furstenberg sets in is no smaller than , where depends only on and . For and , this is an -improvement over a result of Wolff from 1999. The same method also yields an -improvement to Kaufman's projection theorem from 1968. We show that if , and is an analytic set with , then where only depends on and . Here is the orthogonal projection to .

Paper Structure

This paper contains 32 sections, 33 theorems, 256 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Background on Furstenberg sets
  4. $(s,t)$-Furstenberg sets
  5. Background on projections
  6. Connections with other problems in geometric measure theory
  7. Proof strategy
  8. Preliminaries and elementary incidence estimates
  9. Notation
  10. $(\delta,s)$-sets
  11. Duality and $(\delta,s)$-sets of lines
  12. An elementary incidence bound
  13. Discretising the main results
  14. Discretising Furstenberg sets
  15. Discretising projections
  16. ...and 17 more sections

Key Result

Theorem 1.1

For every $s \in (0,1)$ and $t\in (s,2]$, there exists $\epsilon = \epsilon(s,t) > 0$ such that the following holds. Let $K \subset \mathbb{R}^{2}$, let $\mathcal{L}$ be a family of lines with $\dim_{\mathrm{H}} \mathcal{L} \geq t$, and assume that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\

Figures (2)

  • Figure 1: Selecting the interval $(c',d_{k})$.
  • Figure 2: The tube $\mathbf{T}_{0}$ drawn with a green outline, and two squares $Q,Q' \in \mathbf{T}_{0}(\mathcal{Q})$. Part of the set $Z$ is also visible as the union of the small red circles on the lower boundaries of the squares $Q,Q' \in \mathbf{T}_{0}(\mathcal{Q})$.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Definition 2.1: $(\delta,s,C)$-set
  • Definition 2.2: Families of $\delta$-cubes
  • Remark 2.3
  • Proposition 2.4
  • ...and 68 more