Furstenberg--Sárközy theorem over number fields

2025/08/26 by Pandey, Dev Ranjan, Saha, Jyoti Prakash
#05B10 #11B30 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.18990

Abstract

We introduce the notion of intersective polynomials having coefficients in the ring of integers \mathscrO_K of a number field K, and define a notion of upper density of subsets of \mathscrO_K. We prove that given any intersective polynomial p(x) over \mathscrO_K, every subset A of \mathscrO_K of positive upper density contains two distinct elements whose difference is equal to p(x) for some element x in \mathscrO_K. Moreover, we obtain a quantitative version of this result. The proof is motivated by an argument due to Lucier, and the Fourier-free proof of the Furstenberg--Sárközy theorem over the integers by Green, Tao and Ziegler.

Citations

• New bounds for the Furstenberg-Sárközy theorem
• A maximal extension of the Bloom-Maynard bound for sets with no square differences
• A new upper bound for sets with no square differences
• A new upper bound for sets with no square differences
• A Maximal Extension of the Best-Known Bounds for the Furstenberg-Sárközy Theorem
• Improved Bounds on Sarkozy's Theorem for Quadratic Polynomials

Related

• Sarkozy's Theorem for P-Intersective Polynomials
• A Maximal Extension of the Best-Known Bounds for the Furstenberg-Sárközy Theorem
• Polynomial actions of rings of integers of global fields and quasirandomness of Paley-type graphs
• Sárközy's Theorem in Various Finite Field Settings
• Sarkozy's theorem in function fields