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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The results by A.Blokhuis and F.Mazzocca about the finite Kakeya problem in two dimensional case, cited by Simeon Ball, will appear on the last volume of Bolyai Society Mathematical Studies dedicated to L.Lovász’s 60th birthday ( http://www.springer.com/math/numbers/book/978-3-540-85218-6?detailsPage=toc )

Dvir has a new survey article online: http://www.eccc.uni-trier.de/report/2009/077/