Sums and products in sets of positive density

TL;DR

This paper uses Fourier analysis and ergodic theory to prove that sets of positive density contain specific sum-product configurations involving polynomial shifts.

Contribution

It introduces a novel analytic framework to establish sum-product results in sets of positive density with polynomial conditions.

Findings

Sets of positive upper logarithmic density contain sum-product configurations when polynomial conditions are met.

A new multiplicative density notion guarantees the presence of sum-product configurations.

The approach bridges Fourier analysis, ergodic theory, and combinatorial number theory.

Abstract

We develop an analytic approach that draws on tools from Fourier analysis and ergodic theory to study Ramsey-type problems involving sums and products in the integers. Suppose $Q$ denotes a polynomial with integer coefficients. We establish two main results. First, we show that if $Q(1) = 0$, then any set of natural numbers with positive upper logarithmic density contains a pair of the form $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N} \setminus \{1\}$. Second, we prove that if $Q(0) = 0$, then any set of natural numbers with positive density relative to a new multiplicative notion of density, which arises naturally in the context of such problems, contains $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N}$.