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

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 main result of this papers also follows from Corollary 1.10 of \cite{WZ25}, which is in fact a strictly stronger result. We thank Josh Zahl for bringing this to the author's attention. The manuscript will be updated to reflect this soon