Sárközy's theorem in various finite field settings

In this talk, we strengthen a result by Ben Green on an analogue of Sárközy’s theorem in the setting of polynomial rings F_q[x]. In the integer setting, for a given polynomial F ∈ Z[x] with constant term zero, (a generalization of) Sárközy’s theorem gives an upper bound on the maximum size of a subset A ⊂ {1, . . . , n} that does not contain distinct a,b ∈ A satisfying a-b = F(c) for some c ∈ Z. Green proved an analogous result with much stronger bounds in the setting of subsets A ⊂ F_q[x] of the polynomial ring F_q[x], but required the additional condition that the number of roots of the polynomial F ∈ Fq[x] is coprime to q.We generalize Green’s result, removing this condition. As an application, we also show how to obtain a version of Sárközy’s theorem with similarly strong bounds for subsets A ⊂ F_q where q = p^n for a fixed prime p and large n.

This is joint work with Lisa Sauermann.