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# Hunter, Cauchy Rabbit, and Optimal Kakeya Sets

Babichenko, Yakov and Peres, Yuval and Peretz, Ron and Sousi, Perla and Winkler, Peter (2014) Hunter, Cauchy Rabbit, and Optimal Kakeya Sets. Transactions of the American Mathematical Society, 366 (10). pp. 5567-5586. ISSN 0002-9947. doi:10.1090/S0002-9947-2014-06226-0. https://resolver.caltech.edu/CaltechAUTHORS:20140918-150052882

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## 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 Θ(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 ≥ 0} and {W(s) : s ≥ 0}. Let τ_t := min{s ≥ 0 : B(s) = t}. Then X_t = W(τ_t) is a Cauchy process and K := {(ɑ,X_t + ɑt) : ɑ, t ∈ [0, 1]} is a Kakeya set of zero area. The area of the ε-neighbourhood of K is as small as possible, i.e., almost surely of order Θ(1/| log ε|).

Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1090/S0002-9947-2014-06226-0DOIArticle
http://arxiv.org/abs/1207.6389arXivDiscussion Paper
Additional Information:© 2014 American Mathematical Society. Received by the editors February 13, 2013. Article electronically published on June 10, 2014. 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. The authors thank Abraham Neyman for useful discussions.
Funders:
Funding AgencyGrant Number
European Research Council (ERC)039-7679-411
Israel Science Foundation212/09
Google Inter-University Center for Electronic Markets and AuctionsUNSPECIFIED
Microsoft ResearchUNSPECIFIED
MSRI Simons ProfessorshipUNSPECIFIED
NSFDMS-0901475
Subject Keywords:Pursuit games, graph games, Kakeya sets, Cauchy process
Issue or Number:10
Classification Code:MSC (2010): Primary 49N75; Secondary 05C57, 60G50
DOI:10.1090/S0002-9947-2014-06226-0
Record Number:CaltechAUTHORS:20140918-150052882
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20140918-150052882
Official Citation:Hunter, Cauchy rabbit, and optimal Kakeya sets Yakov Babichenko, Yuval Peres, Ron Peretz, Perla Sousi and Peter Winkler. Trans. Amer. Math. Soc. 366 (2014), 5567-5586
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:49838
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:18 Sep 2014 22:35