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), 55675586
MSC (2010):
Primary 49N75; Secondary 05C57, 60G50
Published electronically:
June 10, 2014
MathSciNet review:
3240935
Fulltext PDF
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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 . 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 noniterative construction of a boundaryoptimal 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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 Micah Adler, Harald Räcke, Naveen Sivadasan, Christian Sohler, and Berthold Vöcking, Randomized pursuitevasion in graphs, Combin. Probab. Comput. 12 (2003), no. 3, 225244. Combinatorics, probability and computing (Oberwolfach, 2001). MR 1988975 (2004e:60019), http://dx.doi.org/10.1017/S0963548303005625
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 A. S. Besicovitch, On Kakeya's problem and a similar one, Math. Z. 27 (1928), no. 1, 312320. MR 1544912, http://dx.doi.org/10.1007/BF01171101
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 A. S. Besicovitch, The Kakeya problem, Amer. Math. Monthly 70 (1963), 697706. MR 0157266 (28 #502)
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 J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147187. MR 1097257 (92g:42010), http://dx.doi.org/10.1007/BF01896376
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 Roy O. Davies, Some remarks on the Kakeya problem, Proc. Cambridge Philos. Soc. 69 (1971), 417421. MR 0272988 (42 #7869)
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 Ben Green,
Restriction and Kakeya Phenomena. Lecture notes from a course at Cambridge, https://www.dpmms.cam.ac.uk/bjg23/rkp.html.
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 U. Keich, On bounds for Kakeya maximal functions and the Minkowski dimension in , Bull. London Math. Soc. 31 (1999), no. 2, 213221. MR 1664129 (2000b:42016), http://dx.doi.org/10.1112/S0024609398005372
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 David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson. MR 2466937 (2010c:60209)
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 Peter Mörters and Yuval Peres, Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010. With an appendix by Oded Schramm and Wendelin Werner. MR 2604525 (2011i:60152)
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 Oskar Perron, Über einen Satz von Besicovitsch, Math. Z. 28 (1928), no. 1, 383386 (German). MR 1544966, http://dx.doi.org/10.1007/BF01181172
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 I. J. Schoenberg, On the BesicovitchPerron solution of the Kakeya problem, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 359363. MR 0146698 (26 #4218)
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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:
http://dx.doi.org/10.1090/S000299472014062260
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 0307950411 and ISF 0397679411.
The third author’s work was supported in part by grant #212/09 of the Israel Science Foundation and by the Google Interuniversity 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 DMS0901475
The authors also thank MSRI, Berkeley, where part of this work was completed
Article copyright:
© Copyright 2014
American Mathematical Society
