Tag Archives: geometry

Update on 2-dimensional Kakeya sets over finite fields

A few months ago I wrote about the problem of giving a sharp lower bound for the size of a Kakeya subset of F_q^2; that is, a subset containing a line in every direction. Apparently this problem has now been solved: Simeon Ball, in a comment on Terry’s blog, says that Blokhuis and Mazzocca show that a Kakeya set has have cardinality at least q(q+1)/2 + (q-1)/2. Since there are known examples of Kakeya sets of this size arising from conics, this bound is sharp. The argument appears to be completely combinatorial, not algebro-geometric a la Dvir.

I’m a bit confused about attribution; Ball calls it a result of Blokhuis and Mazzocca, but links to a preprint of his own which apparently proves the theorem, and which doesn’t mention Blokhuis and Mazzocca. So who actually proved this nice result? Anyone with first-hand knowledge should enlighten me in comments.

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