Hunter, Cauchy rabbit, and optimal Kakeya sets

Authors:
Yakov Babichenko, Yuval Peres, Ron Peretz, Perla Sousi and Peter Winkler

Journal:
Trans. Amer. Math. Soc. **366** (2014), 5567-5586

MSC (2010):
Primary 49N75; Secondary 05C57, 60G50

Published electronically:
June 10, 2014

MathSciNet review:
3240935

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Abstract | References | Similar Articles | Additional Information

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 . A hunter and a rabbit move on the nodes of 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 steps of order . We show these strategies yield a Kakeya set consisting of triangles with minimal area (up to constant), namely . 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 and . Let . Then is a Cauchy process and is a Kakeya set of zero area. The area of the -neighbourhood of is as small as possible, i.e., almost surely of order .

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Additional Information

**Yakov Babichenko**

Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125

**Yuval Peres**

Affiliation:
Microsoft Research, Redmond, Washington 98052

**Ron Peretz**

Affiliation:
Department of Mathematics, London School of Economics and Political Sciences, London WC2A 2AE, United Kingdom

**Perla Sousi**

Affiliation:
Department of Pure Mathematics, University of Cambridge, Cambridge CB3 0WB, United Kingdom

**Peter Winkler**

Affiliation:
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755

DOI:
https://doi.org/10.1090/S0002-9947-2014-06226-0

Keywords:
Pursuit games,
graph games,
Kakeya sets,
Cauchy process

Received by editor(s):
February 13, 2013

Published electronically:
June 10, 2014

Additional Notes:
The first author would like to acknowledge financial support by ERC 030-7950-411 and ISF 039-7679-411.

The third author’s work was supported in part by grant #212/09 of the Israel Science Foundation and by the Google Inter-university center for Electronic Markets and Auctions

The fifth author’s work was supported by Microsoft Research, by a Simons Professorship at MSRI, and by NSF grant DMS-0901475

The authors also thank MSRI, Berkeley, where part of this work was completed

Article copyright:
© Copyright 2014
American Mathematical Society