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Infinite linear patterns in sets of positive density

Felipe Hernández

TL;DR

The paper addresses identifying all infinite linear configurations that must appear in shifts of sets with positive upper Banach density. It develops an ergodic-theoretic framework based on a progressive measure and pronilfactors, and proves a general theorem for admissible linear forms that subsumes Szemerédi-type results and the Kra–Moreira–Richter–Robertson program. The key technical tool, Lemma-new-method, links positivity of certain sigma-integrals to positive density patterns via nilsystems and uniform Szemerédi theorems. This work broadens the landscape of combinatorial density patterns, providing a unified method to detect complex linear configurations in dense sets with potential applications in additive combinatorics and ergodic theory.

Abstract

In this article we describe all possible infinite linear configurations that can be found in a shift of any set of positive upper Banach density. This simultaneously generalizes Szemerédi's theorem on arithmetic progressions and the recent density finite sums theorem of Kra, Moreira, Richter, and Robertson.

Infinite linear patterns in sets of positive density

TL;DR

The paper addresses identifying all infinite linear configurations that must appear in shifts of sets with positive upper Banach density. It develops an ergodic-theoretic framework based on a progressive measure and pronilfactors, and proves a general theorem for admissible linear forms that subsumes Szemerédi-type results and the Kra–Moreira–Richter–Robertson program. The key technical tool, Lemma-new-method, links positivity of certain sigma-integrals to positive density patterns via nilsystems and uniform Szemerédi theorems. This work broadens the landscape of combinatorial density patterns, providing a unified method to detect complex linear configurations in dense sets with potential applications in additive combinatorics and ergodic theory.

Abstract

In this article we describe all possible infinite linear configurations that can be found in a shift of any set of positive upper Banach density. This simultaneously generalizes Szemerédi's theorem on arithmetic progressions and the recent density finite sums theorem of Kra, Moreira, Richter, and Robertson.

Paper Structure

This paper contains 7 sections, 13 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Basic notions in Ergodic Theory
  5. The progressive measure
  6. The main result
  7. Proving \ref{['Lemma-new-method']}

Key Result

Theorem 1.1

If $A\subseteq \mathbb{N}$ has positive upper Banach density then for every $d\in \mathbb{N}$ there is an integer $t\geq 0$ and an infinite set $B \subseteq \mathbb{N}$ such that

Theorems & Definitions (26)

  • Theorem 1.1: kmrr25
  • Theorem 1.2
  • Theorem 1.3: Reformulation of \ref{['Conjecture-1']} using linear forms
  • Example 1.4
  • Example 1.5
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • Theorem 3.4: Uniform version of Szemerédi's theorem
  • ...and 16 more