Hausdorff dimension of restricted Kakeya sets

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.

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