Sárközy's theorem in \mathbbF_q[t] via the van der Corput property

2025/10/31 by Steve Fan, Fan, Steve, Andrew Lott +1
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #Limits and Structures in Graph Theory #math.NT

paper · pdf · doi:10.48550/arxiv.2510.27581

openalex publication_date 2025/10/31 · openalex created_date 2025/11/05 · openalex updated_date 2026/08/01

Abstract

Fix a positive prime power q, and let \mathbbF_q[t] be the ring of polynomials over the finite field \mathbbF_q with char(\mathbbF_q)>2. Suppose A ⊆ \f ∈ \mathbbF_q[t]: deg f ≤ N\ contains no pair of elements whose difference is of the form P-1 with P irreducible. Adapting Green's approach to Sárközy's theorem for shifted primes in ℤ using the van der Corput property, we show that |A| ≪ q^{(N+1)(11/12+o(1))}, improving upon the bound O(q^{(1-c/log N)(N+1)}) due to Lê and Spencer. An important distinction between Green's argument and ours lies in the properties of exponential sums over function fields, which differ in several interesting ways from their number-field counterparts.

Citations

• On Sárközy's theorem for shifted primes
• Sarkozy's theorem in function fields
• The variance of the number of prime polynomials in short intervals and in residue classes
• Difference sets and shifted primes

Related

• Sárközy's Theorem in Various Finite Field Settings
• Sárközy's theorem in F₂[x]
• Sarkozy's theorem in function fields
• Sarkozy's Theorem for P-Intersective Polynomials
• On Sárközy's theorem for shifted primes