Heisenberg Hausdorff dimension of Besicovitch sets

TL;DR

This paper explores the relationship between Kakeya maximal function estimates and the Heisenberg Hausdorff dimension of Besicovitch sets within the Heisenberg group, establishing new lower bounds.

Contribution

It demonstrates that $L^p$ bounds for the Kakeya maximal function lead to lower bounds on the Heisenberg Hausdorff dimension of Besicovitch sets, linking harmonic analysis to geometric measure theory.

Findings

$L^p$ estimates imply lower bounds for Heisenberg Hausdorff dimension

Establishes a connection between Kakeya maximal function bounds and geometric properties

Provides new insights into the structure of Besicovitch sets in the Heisenberg group

Abstract

We consider (bounded) Besicovitch sets in the Heisenberg group and prove that $L^p$ estimates for the Kakeya maximal function imply lower bounds for their Heisenberg Hausdorff dimension.