Sárközy’s Theorem in Various Finite Field Settings

.In this paper, we strengthen a result by Green about 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 \(a_1,a_2 ∈ A\) satisfying \(a_1 - a_2 = F(b)\) for some \( b ∈ Z\). 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 this result required the additional condition that the number of roots of the polynomial \(F ∈ F_q[x]\) be coprime to \(q\). We generalize Green's result, removing this condition. As an application, we also obtain a version of Sárközy's theorem with similar strong bounds for subsets \(A ⊂ F_q\) for \(q = p^n\) for a fixed prime \(p\) and large \(n\).