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 $2t+1$.