The Kakeya Set conjecture over Z mod N for general N

A Kakeya Set in (Z/N Z)^n is a set that contains a line in every direction.  It has been known for over a decade that such sets must be large when N is prime (or more generally over any finite field).  This goes back to Dvir's proof of the finite field Kakeya conjecture as posed by Wolff. In this talk I will cover results from three papers from the past ~year culminating in the resolution of the conjecture for any N, showing that, for every eps>0, a Kakeya set S in (Z/N Z)^n must have size at least C*N^{n-eps} where the constant C depends on n and eps only. The first paper (by the speaker and Dvir) resolved the case of square-free N.  The second paper (by Arsovski) settled the prime power case (which implies the p-adic Kakeya conjecture for Hausdorff dimension).  Finally, a paper by the speaker, building on ideas from both prior works, proves the conjecture for any N.  The proofs involve several extensions/modifications to the polynomial method (as used by Dvir for the finite field case) which may be of independent interest.



Manik Dhar


Princeton University