Almost all non-archimedean Kakeya sets have measure zero
Confluentes Mathematici, Volume 10 (2018) no. 1, pp. 3-40.

We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets are neglectable according to the Haar measure. We also discuss possible relations with the non-archimedean Kakeya conjecture.

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DOI: 10.5802/cml.44
Classification: 05B30, 51E20, 60B11, 11K41
Keywords: Kakeya set, discrete valuation fields
Xavier Caruso 1

1 Université Rennes 1, IRMAR, Campus de Beaulieu, 35042 Rennes Cedex
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Xavier Caruso. Almost all non-archimedean Kakeya sets have measure zero. Confluentes Mathematici, Volume 10 (2018) no. 1, pp. 3-40. doi : 10.5802/cml.44. https://cml.centre-mersenne.org/articles/10.5802/cml.44/

[1] Y. Babichenko, Y. Peres, R. Peretz, P. Sousi, P. Winkler, Hunter, Cauchy Rabbit, and Optimal Kakeya Sets, Trans. Amer. Math. Soc. 366 (2014), 5567–5586 | DOI | MR | Zbl

[2] A. Besicovitch, On Kakeya’s problem and a similar one, Math. Z. 27 (1928), 312–320 | DOI | MR | Zbl

[3] A. Cordoba, The Kakeya maximal function and the spherical summation multiplier, Amer. J. Math. 99 (1971), 1–22 | DOI | MR | Zbl

[4] R. Davies, Some remarks on the Kakeya problem, Proc. Cambridge Philos. Soc. 69 (1971), 417–421 | DOI | MR | Zbl

[5] E. Dummit, M. Hablicsek, Kakeya sets over non-archimedean local rings, Mathematika 59 (2013), 257–266 | DOI | MR | Zbl

[6] Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), 1093–1097 | DOI | MR | Zbl

[7] J. Ellenberg, R. Oberlin, T. Tao, The Kakeya set and maximal conjectures for algebraic varieties over finite fields, Mathematika 56 (2010), 1–25 | DOI | MR | Zbl

[8] C. Fefferman, The multiplier problem for the ball, Ann. of Math. 94 (1971), 330–336 | DOI | MR | Zbl

[9] R. Fraser, Kakeya-Type Sets in Local Fields with Finite Residue Field, Mathematika 62 (2016), 614–629 | DOI | MR | Zbl

[10] L. Guth, A short proof of multilinear Kakeya inequality, Math. Proc. Cambridge Philos. Soc. 158 (2015), 147–153 | DOI | MR | Zbl

[11] L. Guth, J. Zahl, Polynomial Wolff axiomes and Kakeya-type estimates in 4 , preprint (2017), available at | arXiv | DOI | Zbl

[12] B. Green, Restriction and Kakeya Phenomena, lecture notes from a course at Cambridge, http://people.maths.ox.ac.uk/greenbj/papers/rkp.pdf

[13] N. Katz, J. Zahl, An improved bound on the Hausdorff dimension of Besicovitch sets in 3 , preprint (2017), available at | arXiv | DOI | MR

[14] S. Kakeya, Some problems on maximum and minimum regarding ovals, Tohoku science reports 6 (1917), 71–88 | Zbl

[15] N. Katz, I. Łaba, T. Tao, An improved bound for the Minkowski dimension of Besikovitch sets in 3 , Annals of Math. 152 (2000), 383–446 | DOI | MR | Zbl

[16] N. Katz, T. Tao, New bounds for Kakeya problems, J. Anal. Math. 87 (2002), 231–263 | DOI | MR | Zbl

[17] N. Katz, T. Tao, Recent progress on the Kakeya conjecture, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), Publ. Mat. (2002), 161–179. | DOI | MR

[18] I. Łaba, T. Tao, An improved bound for the Minkowski dimension of Besikovitch sets in medium dimension, Geom. Funct. Anal. 11 (2001), 773–806 | DOI | MR | Zbl

[19] T. Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), 651–674 | DOI | MR | Zbl

[20] T. Wolff, Recent work connected with the Kakeya problem, in Prospects in mathematics (Princeton, NJ, 1996), pp. 129–162, Amer. Math. Soc., Providence, RI (1999) | DOI | Zbl

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