Hunter, Cauchy Rabbit, and Optimal Kakeya Sets

TL;DR

This paper connects pursuit game strategies on cycle graphs to the construction of minimal-area Kakeya sets, providing the first non-iterative boundary-optimal example and a continuum analog using Brownian motions and Cauchy processes.

Contribution

It introduces a novel non-iterative construction of boundary-optimal Kakeya sets based on pursuit game strategies and extends the concept to a continuum setting with Brownian motions and Cauchy processes.

Findings

Constructs Kakeya sets of minimal area using pursuit game strategies.

Provides a continuum analog with Brownian motions leading to a zero-area Kakeya set.

Shows the area of the epsilon-neighborhood of the set is minimal, of order 1/|log epsilon|.

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 $\Z_n$. A hunter and a rabbit move on the nodes of $\Z_n$ 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 $\epsilon$-neighborhood of $K$ is as small as possible, i.e., almost surely of order $\Theta(1/|\log \epsilon|)$.