Polynomial configurations in dense subsets of the prime lattice

2025/04/19 by Lott, Andrew, Magyar, Ákos, Petridis, Giorgis +1
#11N05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.14424

Abstract

We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let A be a subset of the prime lattice - the d-fold direct product of the primes - of positive relative upper density. We show that A contains all polynomial configurations of the form x+P₀(y)v₀,…, x+P_l(y)v_l, for some x in ℤ^d and y in ℕ, which satisfy a certain non-degeneracy condition. We also obtain quantitative bounds on the size of such polynomial configuration, if A is a subset of the first N positive integers.

Related

• Narrow progressions in the primes
• Multidimensional configurations in the primes with shifted prime steps
• Random linear configurations in dense sets and primes
• Small gaps between configurations of prime polynomials
• Sarkozy's Theorem for P-Intersective Polynomials