A new lower bound for Szemerédi's theorem with random differences in finite fields

For what size random subset D of F_p^n does it hold, with high probability, that any dense subset contains a nontrivial k-term arithmetic progression with common difference in D? We provide a new lower bound on the size of D by showing that a sufficiently small D will be disjoint from a dense algebraic set with high probability. In particular we obtain a leading constant which grows as the threshold for what is considered a "dense" set in Szemerédi's theorem shrinks. 

Our general result is a sharp threshold for the following problem: how many points in F_p^n does one need to randomly sample to almost surely intersect every algebraic set defined by at most s polynomials each of degree at most k? 

Joint with Daniel Altman