Sárközy's theorem in F₂[x]

2025/10/31 by Kowalska, Aleksandra
Mathematics · #11B30 (Primary) 11P70 #11P55 (Secondary) #11T55 #Algebraic Geometry and Number Theory #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.2510.27573

openalex publication_date 2025/10/31 · openalex created_date 2025/11/05 · openalex updated_date 2026/07/28

Abstract

Green showed that, conditional on GRH, a subset A ⊆ [N] with | A | ≫_ε N^(11)/(12)+ε must contain two elements whose difference is p-1 for p a prime. We prove an analogous unconditional result for F₂[x], improving the exponent to (7)/(8)+ε.

Citations

• On Sárközy's theorem for shifted primes
• Counting irreducible polynomials over finite fields using the inclusion-exclusion principle

Related

• Sárközy's theorem in \mathbbF_q[t] via the van der Corput property
• Sarkozy's Theorem for P-Intersective Polynomials
• Sárközy's Theorem in Various Finite Field Settings
• On Sárközy's theorem for shifted primes
• The Furstenberg-Sárközy theorem for polynomials in one or more prime variables