Archiv der Mathematik

, Volume 103, Issue 3, pp 267–277 | Cite as

Variants of the Kakeya problem over an algebraically closed field



First, we study constructible subsets of \({\mathbb{A}^n_k}\) which contain a line in any direction. We classify the smallest such subsets in \({\mathbb{A}^3}\) of the type \({R \cup \{g \neq 0\},}\) where \({g \in k[x_1,\ldots, x_n]}\) is irreducible of degree d and \({R \subset V(g)}\) is closed. Next, we study subvarieties \({X \subset \mathbb{A}^N}\) for which the set of directions of lines contained in X has the maximal possible dimension. These are variants of the Kakeya problem in an algebraic geometry context.


Kakeya problem Constructible sets Bezout’s theorem Intersection multiplicity Flexy surface Funny curve Projective cone Fano variety 

Mathematics Subject Classification

14R05 14N05 14N15 14N20 05B05 


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  1. 1.
    Dummit E., Hablicsek M.: Kakeya sets over non-archimedean local rings. Mathematika 59, 257–266 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Dvir Z.: On the size of Kakeya sets in finite fields. J. Amer. Math. Soc. 22, 1093–1097 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Rogora E.: Varieties with many lines. Manuscripta Math. 82, 207–226 (1994)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    S. Saraf and M. Sudan, Improved lower bound on the size of Kakeya sets over finite fields, Anal. PDE 1 (2008), 375–379.Google Scholar
  5. 5.
    K. Slavov, An algebraic geometry version of the Kakeya problem, in preparation.Google Scholar
  6. 6.
    Wolff T.: An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoamericana 11, 651–674 (1999)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.American University in BulgariaBlagoevgradBulgaria

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