On the size of Kakeya sets in finite fields

By Zeev Dvir. 2008

283 citations recorded by the source corpus.

The motivation for studying Kakeya sets over finite fields is to try to better understand the more complicated questions regarding Kakeya sets in W1. A Kakeya set K C Rn is a compact set containing a line segment of unit length in every direction. The famous Kakeya Conjecture states that such sets must have Hausdorff (or Minkowski) dimension equal to n. The importance of this conjecture is partially due to the connections it has to many problems in harmonic analysis, number theory and PDE. This conjecture was proved for n = 2 [Dav71] and is open for larger values of n (we refer the reader to the survey papers [Wol99, BouOO, TaoOl] for more information). It was first suggested by Wolff [Wol99] to study finite field Kakeya sets. It was asked in [Wol99] whether there exists a lower bound of the form Cn • qn on the size of such sets in Fn. The lower bound appearing in [Wol99] was of the form Cn c(n+2)/2. This bound was further improved in [RogOl, BKT04, MT04, Tao08] both for general n and for specific small values of n (e.g. for n = 3, 4). For general n, the most current best lower bound is the one obtained in [RogOl, MT04] (based on results from [KT99]) of Cn • q4n/7 . The main technique used to show this bound is an additive number theoretic lemma relating the sizes of different sum sets of the form A+rB, where A and B are fixed sets in Fn and r ranges over several different values in F (the idea to use additive number theory in the context of Kakeya sets is due to Bourgain [Bou99]). The next theorem, proven in Section 2, gives a near-optimal bound on the size of Kakeya sets. Roughly speaking, the proof follows by observing that any degree q 2 homogeneous polynomial in F[#i, . . . , xn] can be 'reconstructed' from its value on any Kakeya set K c¥n. This implies that the size of K is at least the dimension of the space of polynomials of degree q 2, which is « q71'1 (when q is large).

Subfields: Applied Mathematics, Artificial Intelligence, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/a6eee75dc0780e1d3cd05e61ae242e1721fe9bbc

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Citations in the corpus

Algebraic methods in discrete analogs of the Kakeya problem (2008, methodology): “Both our proofs are adaptations of Dvir’s argument for the finite field Kakeya problem.”
Modular hyperbolas (2011, background): “1] and also surveys [32, 120] (we also note the recent remarkable achievement [69, 105]).”
New Restriction Estimates for the 3-d Paraboloid over Finite Fields (2013, background): “Indeed, in 2008 Dvir [4] proved the full finite field Kakeya conjecture, however neither his result nor its proof technique have yet yielded any progress on the finite field restriction problem.”
Improved lower bound on the size of Kakeya sets over finite fields (2008, methodology): “Proposition 6 ([1]) Let g ∈ F[x] be a non-zero polynomial of total degree d and let g0 be the (unique, non-zero) homogenous polynomial of degree d such that g = g0 + g1 for some polynomial g1 of degree strictly less than d.”