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 \ (Fqx\). In the integer setting, for a given polynomial \ (F Zx\) 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 \ (a₁, a₂ A\) satisfying \ (a₁ - a₂ = F (b) \) for some \ (b Z\). Green proved an analogous result with much stronger bounds in the setting of subsets \ (A Fqx\) of the polynomial ring \ (Fqx\), but this result required the additional condition that the number of roots of the polynomial \ (F Fqx\) 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 Fq\) for \ (q = pⁿ\) for a fixed prime \ (p\) and large \ (n\). Keywordspolynomial methodadditive combinatoricsSárközy's theoremnumber theorycombinatoricsMSC codes05D4005D99