Polynomial multiple recurrence over rings of integers

TL;DR

This paper extends the polynomial Szemerédi theorem to algebraic number fields, establishing new polynomial configurations in dense subsets of integer lattices and broadening previous integer-based results.

Contribution

It generalizes polynomial recurrence results to rings of integers in algebraic number fields, introducing intersective polynomials with common roots modulo ideals.

Findings

Existence of polynomial configurations in positive-density subsets of b2^m

Extension of results from integers to algebraic number fields

Strengthening of previous polynomial recurrence theorems

Abstract

We generalize the polynomial Szemer\'{e}di theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new polynomial configurations in positive-density subsets of $\mathbb{Z}^m$ and strengthens and extends recent results of Bergelson, Leibman and Lesigne on polynomials over the integers.