Sárközy’s Theorem in Various Finite Field Settings

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1409-1416, June 2024.
Abstract. In this paper, we strengthen a result by Green about an analogue of Sárközy’s theorem in the setting of polynomial rings [math]. In the integer setting, for a given polynomial [math] with constant term zero, (a generalization of) Sárközy’s theorem gives an upper bound on the maximum size of a subset [math] that does not contain distinct [math] satisfying [math] for some [math]. Green proved an analogous result with much stronger bounds in the setting of subsets [math] of the polynomial ring [math], but this result required the additional condition that the number of roots of the polynomial [math] be coprime to [math]. 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 [math] for [math] for a fixed prime [math] and large [math].