Hunter, Cauchy rabbit, and optimal Kakeya sets

Abstract:

A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle $\mathbb {Z}_n$. A hunter and a rabbit move on the nodes of $\mathbb {Z}_n$ without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al. (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first $n$ steps of order $1/\log n$. We show these strategies yield a Kakeya set consisting of $4n$ triangles with minimal area (up to constant), namely $\Theta (1/\log n)$. As far as we know, this is the first non-iterative construction of a boundary-optimal Kakeya set. Considering the continuum analog of the game yields a construction of a random Kakeya set from two independent standard Brownian motions $\{B(s): s \ge 0\}$ and $\{W(s): s \ge 0\}$. Let $\tau _t:=\min \{s \ge 0: B(s)=t\}$. Then $X_t=W(\tau _t)$ is a Cauchy process and $K:=\{(a,X_t+at) : a,t \in [0,1]\}$ is a Kakeya set of zero area. The area of the $\varepsilon$-neighbourhood of $K$ is as small as possible, i.e., almost surely of order $\Theta (1/|\log \varepsilon |)$.