Sums and products in sets of positive density

TL;DR

This work develops an analytic framework blending Fourier analysis and ergodic theory to tackle sum–product Ramsey problems in the integers. It introduces a density notion and a structure theorem that decomposes functions into a structured, locally rationally almost periodic part and a random, locally totally ergodic part, enabling precise correlation estimates for the pattern \{x+Q(y), xy\}. The main results show that when \(Q(1)=0\) positive upper logarithmic density guarantees the existence of the pattern, and when \(Q(0)=0\) a new multiplicative density yields the same conclusion, connecting density notions to Hindman-type phenomena in \(\mathbb{N}\). The paper also develops a multiplicative van der Corput-type inequality and spectral-characterization tools, offering a robust framework for future density versions and extensions of sum–product problems in the integers.

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 $x, y \in \mathbb{N}$.

Theorems & Definitions (51)

• Conjecture 1.1: Hindman's conjecture
• Conjecture 1.2: Generalized Hindman's conjecture
• Conjecture 1.3: Moreira's Conjecture
• Theorem 1.5
• Example 1.6
• Theorem 1.7
• Example 1.8
• Definition 1.9
• Theorem 1.10
• proof : Proof that \ref{['thm_1_tech']} implies \ref{['thm_1']}