New bounds for Szemeredi's Theorem, I: Progressions of length 4 in finite field geometries

Let F be a fixed finite field of characteristic at least 5. Let G = F^n be the n-dimensional vector space over F, and write N := |G|. We show that if A is a subset of G with size at least c_F N(log N)^{-c}, for some absolute constant c > 0 and some c_F > 0, then A contains four distinct elements in arithmetic progression. This is equivalent, in the usual notation of additive combinatorics, to the assertion that r_4(G) <<_F N(log N)^{-c}.