2022/12/24 by Anqi Li, Li, Anqi, Lisa Sauermann +1 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2212.12754
openalex publication_date 2022/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper, we strengthen a result by Green about an analogue of Sarkozy's theorem in the setting of polynomial rings \mathbbFq[x]. In the integer setting, for a given polynomial F ∈ ℤ[x] with constant term zero, (a generalization of) Sarkozy'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 ⊂ \mathbbFq[x] of the polynomial ring \mathbbFq[x], but required the additional condition that the number of roots of the polynomial F ∈ \mathbbFq[x] is coprime to q. We generalize Green's result, removing this condition. As an application, we also obtain a version of Sarkozy's theorem with similarly strong bounds for subsets A ⊂ \mathbbFq for q = pn for a fixed prime p and large n.