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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Hunter, Cauchy rabbit, and optimal Kakeya sets
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by Yakov Babichenko, Yuval Peres, Ron Peretz, Perla Sousi and Peter Winkler PDF
Trans. Amer. Math. Soc. 366 (2014), 5567-5586 Request permission

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 $\Theta (1/|\log \varepsilon |)$.
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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
  • MR Author ID: 137920
  • 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
  • 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
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 5567-5586
  • MSC (2010): Primary 49N75; Secondary 05C57, 60G50
  • DOI: https://doi.org/10.1090/S0002-9947-2014-06226-0
  • MathSciNet review: 3240935