Classical combinatorial geometric theorems over Euclidean spaces R^n do not hold over finite fields. For instance, the Heisenberg surface that is cut out by the equation x-x^p+y^pz-yz^p=0 from the three dimensional space over the field with p^2 elements contains few number of points relative to the number of lines it contains. The Kakeya problem gives another example for this phenomenon: Dvir showed that over finite fields the measure of a Kakeya set is always “big”. Following Ellenberg, Oberlin and Tao we consider the Kakeya problem in a new setting, over non-archimedean rings. This work is joint with Evan Dummit.
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