Infinite polynomial patterns in large subsets of the rational numbers

TL;DR

This paper demonstrates that large subsets of the rational numbers contain complex infinite polynomial configurations that are guaranteed by Ramsey theory and density arguments, unlike in the integers.

Contribution

It introduces new combinatorial results about polynomial configurations in rational numbers, contrasting their properties with integers, and develops novel dynamical tools for analysis.

Findings

Existence of monochromatic infinite polynomial configurations in rational colorings

Presence of such configurations in positive density subsets of rationals

Differences in combinatorial structures between rationals and integers

Abstract

Inspired by a question of Kra, Moreira, Richter, and Robertson, we prove two new results about infinite polynomial configurations in large subsets of the rational numbers. First, given a finite coloring of $\mathbb{Q}$, we show that there exists an infinite set $B = \{b_n : n \in \mathbb{N}\} \subseteq \mathbb{Q}$ such that $$\{b_i, b_i^2 + b_j : i < j\}$$ is monochromatic. Second, we prove that every subset of positive density in the rational numbers contains a translate of such an infinite configuration. The corresponding results in the integers are both known to be false, so our results provide natural and relatively simple examples of combinatorial structures that distinguish between the Ramsey-theoretic properties of the rational numbers and the integers. The proofs of our main results build upon methods developed in a series of papers by Kra, Moreira, Richter, and Robertson to translate from combinatorics into dynamics, where the core of the argument reduces to understanding the behavior of certain polynomial ergodic averages. The new dynamical tools required for this analysis are a Wiener--Wintner theorem for polynomially-twisted ergodic averages in $\mathbb{Q}$-systems and a structure theorem for Abramov $\mathbb{Q}$-systems.