Hausdorff dimension of restricted Kakeya sets

TL;DR

This paper investigates the Hausdorff dimension of restricted Kakeya sets where midpoints lie in a set with limited packing dimension, establishing new lower bounds and extending results to Kakeya maximal functions.

Contribution

It introduces improved lower bounds for the Hausdorff dimension of restricted Kakeya sets using the bush argument and extends these bounds to Kakeya maximal function analogues.

Findings

Hausdorff dimension of restricted Kakeya sets is at least n - s

Improved lower bounds using the bush argument, including max{n - s, n - g_n(s)}

Extension of results to Kakeya maximal function analogues

Abstract

A Kakeya set in $\mathbb{R}^n$ is a compact set that contains a unit line segment $I_e$ in each direction $e \in S^{n-1}$. The Kakeya conjecture states that any Kakeya set in $\mathbb{R}^n$ has Hausdorff dimension $n$. We consider a restricted case where the midpoint of each line segment $I_e$ must belong to a fixed set $A$ with packing dimension at most $s \in [0, n]$. In this case, we show that the Hausdorff dimension of the Kakeya set is at least $n - s$. Furthermore, using the "bush argument", we improve the lower bound to $\max \{ n - s, n - g_n(s)\}$, where $g_n(s)$ is defined inductively. For example, when $n = 4$, we prove that the Hausdorff dimension is at least $\max\{\frac{19}{5} - \frac{3}{5}s,4-s\}$. We also establish Kakeya maximal function analogues of these results.