Weakly mixing sets and polynomial equations

TL;DR

This paper demonstrates that weakly mixing sets contain polynomial patterns, specifically showing that large integers can be expressed through sums involving polynomial values and elements of the set.

Contribution

It establishes the presence of polynomial configurations in weakly mixing sets, extending understanding of their combinatorial structure.

Findings

Weakly mixing sets contain polynomial patterns for odd-degree polynomials.

Large integers can be represented as sums involving polynomial values and set elements.

The results connect ergodic theory with additive combinatorics in weakly mixing contexts.

Abstract

We investigate polynomial patterns which can be guaranteed to appear in \emph{weakly mixing} sets introduced by introduced by Furstenberg and studied by Fish. In particular, we prove that if $A \subset \mathbb N$ is a weakly mixing set and $p(x) \in \mathbb Z[x]$ a polynomial of odd degree with positive leading coefficient, then all sufficiently large integers $N$ can be represented as $N = n_1 + n_2$, where $p(n_1) + m,\ p(n_2) + m \in A$ for some $m \in A$.