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The quasi-Assouad dimension of $(1,2t)$-Furstenberg sets in $\mathbb{R}^3$ is extremized by sticky sets

Sam Craig

TL;DR

This work proves that sticky $(1,2t)$-Furstenberg sets in $\mathbb{R}^3$ minimize the quasi-Assouad dimension among all $(1,2t)$-Furstenberg sets, by developing a multiscale, prism-based framework anchored in the continuous $t$-Frostman Convex Wolff Axiom. The method propagates lower bounds from self-similar (sticky) configurations to general sets via a detailed discretization, a four-way prism decomposition, and a sequence of induction-on-scales arguments culminating in a $2t+1$-type upper-spectrum bound. A core component is translating stickiness into grain-like structures whose incidence geometry is controlled through Cordoba-type estimates and brush arguments, enabling the transfer of extremal bounds. A key corollary from Wang–Zahl ensures that all $(1,2t)$-Furstenberg sets in dimension three have Hausdorff dimension $2t+1$, linking the quasi-Assouad analysis to classical dimension theory and Kakeya-type problems. Overall, the paper reduces the three-dimensional Furstenberg problem to structural analysis of sticky grain configurations and provides a quantitative, scale-aware extremization principle for the quasi-Assouad dimension in this setting.

Abstract

A $(1,2t)$-Furstenberg set in $\mathbb{R}^3$ is naturally defined as a set containing a union of unit line segments forming a $2t$-dimensional subset of the affine Grassmannian in $\mathbb{R}^3$ and satisfying a suitable variant of the Frostman Convex Wolff Axiom. Some of these sets have a multi-scale self-similarity property called stickiness. We investigate the extremizers of the quasi-Assouad dimension of $(1,2t)$-Furstenberg sets, a slightly stronger variant of the Assouad dimension. We prove that sticky $(1,2t)$-Furstenberg sets have the least possible quasi-Assouad dimension among all $(1,2t)$-Furstenberg sets. This result also follows from Corollary 1.10 of Wang and Zahl's solution to the Kakeya conjecture, which implies that all $(1,2t)$-Furstenberg sets have Hausdorff dimension $2t+1$.

The quasi-Assouad dimension of $(1,2t)$-Furstenberg sets in $\mathbb{R}^3$ is extremized by sticky sets

TL;DR

This work proves that sticky -Furstenberg sets in minimize the quasi-Assouad dimension among all -Furstenberg sets, by developing a multiscale, prism-based framework anchored in the continuous -Frostman Convex Wolff Axiom. The method propagates lower bounds from self-similar (sticky) configurations to general sets via a detailed discretization, a four-way prism decomposition, and a sequence of induction-on-scales arguments culminating in a -type upper-spectrum bound. A core component is translating stickiness into grain-like structures whose incidence geometry is controlled through Cordoba-type estimates and brush arguments, enabling the transfer of extremal bounds. A key corollary from Wang–Zahl ensures that all -Furstenberg sets in dimension three have Hausdorff dimension , linking the quasi-Assouad analysis to classical dimension theory and Kakeya-type problems. Overall, the paper reduces the three-dimensional Furstenberg problem to structural analysis of sticky grain configurations and provides a quantitative, scale-aware extremization principle for the quasi-Assouad dimension in this setting.

Abstract

A -Furstenberg set in is naturally defined as a set containing a union of unit line segments forming a -dimensional subset of the affine Grassmannian in and satisfying a suitable variant of the Frostman Convex Wolff Axiom. Some of these sets have a multi-scale self-similarity property called stickiness. We investigate the extremizers of the quasi-Assouad dimension of -Furstenberg sets, a slightly stronger variant of the Assouad dimension. We prove that sticky -Furstenberg sets have the least possible quasi-Assouad dimension among all -Furstenberg sets. This result also follows from Corollary 1.10 of Wang and Zahl's solution to the Kakeya conjecture, which implies that all -Furstenberg sets have Hausdorff dimension .

Paper Structure

This paper contains 25 sections, 31 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. History of the Furstenberg problem
  3. Background on non-concentration conditions for line segments
  4. Background on the quasi-Assouad dimension
  5. Statement of the main result
  6. Paper outline
  7. Notation
  8. Acknowledgement
  9. Outline of the proof of Theorem 1
  10. Proving Theorem \ref{['thm:realmain']} in the $t=1$ case
  11. Proving Theorem \ref{['thm:realmain']} in the general case: reducing to tubes concentrating in prisms
  12. Proposition \ref{['prop:prismsetup']}: Upper spectrum bouds for tubes concentrating in prisms
  13. Proposition \ref{['prop:smallprism']}: Improving dimensionality of tubes concentrating in prisms
  14. Discretizing the main theorem
  15. A multiscale decomposition for unions of tubes
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose every approximately sticky $(1,2t)$-Furstenberg set has quasi-Assouad dimension $\ge \sigma$. Then every $(1,2t)$-Furstenberg set has quasi-Assouad dimension $\ge \sigma$.

Theorems & Definitions (74)

  • Theorem : 1 (informal version)
  • Corollary : (Theorem 1 informal corollary)
  • Definition
  • Definition
  • Definition 1.1
  • Remark
  • Definition 1.2
  • Definition 1.3
  • Definition
  • Definition 1.4
  • ...and 64 more