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On density analogs of Hindman's finite sums theorem

Felipe Hernández, Ioannis Kousek, Tristán Radić

TL;DR

The paper advances Hindman-type density theorems by proving two density-analog variants: one with a single infinite set $B$ and two growing summand counts controlled by shifts $(t_k)$ and $(s_k)$, and another where summands come from a fixed infinite set $B$ with expanding cardinalities. The approach recasts combinatorial statements into ergodic-theoretic ones using Furstenberg's correspondence principle and introduces progressive and multiple right-progressive measures on infinite product spaces built from the infinite-step pronilfactor $Z_ abla$, enabling inductive constructions of unbounded sumset patterns. Key contributions include optimality results showing the necessity of shifts and the non-extendability to bounded or nested configurations, along with corollaries yielding sumsets $B_1+ obreak obreak+ obreak B_k subseteq A$ or $B_1+ obreak obreak+ obreak B_k eq ext{ bounded in }k$ under certain density constraints. The work thereby strengthens the density version of Hindman’s theorem and provides a robust dynamical framework for density-analog sumset patterns with unbounded summand counts. The methods have potential implications for understanding higher-order combinatorial structures in dense sets via infinite-step nilsystems and pronilfactored dynamics.

Abstract

For any set $A$ of natural numbers with positive upper Banach density, we show the existence of an infinite set $B$ and sequences $(t_k)_{k\in \mathbb{N}}, (s_k)_{k\in \mathbb{N}}$ of natural numbers such that $\left\{ \sum_{n \in F}n : F \subset B + s_k, 1 \leq |F| \leq k \right\}\subset A-t_k$, for every $k\in \mathbb{N}$. This strengthens the density finite sums theorem of Kra, Moreira, Richter, and Robertson. We further show, given such a set $A$, the existence of an infinite set $B$ and a sequence $(t_k)_{k\in \mathbb{N}}$ of natural numbers such that $\left\{ \sum_{n \in F}n : F \subset B, |F| = k \right\}\subset A-t_k$, for every $k\in \mathbb{N}$. As a corollary, we obtain a sequence $(B_n)_{n\in \mathbb{N}}$ of infinite sets of natural numbers such that $B_1+\cdots +B_k \subset A$, for every $k\in \mathbb{N}$. We also establish the optimality of our main theorems by providing counterexamples to potential further generalizations, and thereby addressing questions of the aforementioned authors in the context of density analogs to Hindman's finite sums theorem.

On density analogs of Hindman's finite sums theorem

TL;DR

The paper advances Hindman-type density theorems by proving two density-analog variants: one with a single infinite set and two growing summand counts controlled by shifts and , and another where summands come from a fixed infinite set with expanding cardinalities. The approach recasts combinatorial statements into ergodic-theoretic ones using Furstenberg's correspondence principle and introduces progressive and multiple right-progressive measures on infinite product spaces built from the infinite-step pronilfactor , enabling inductive constructions of unbounded sumset patterns. Key contributions include optimality results showing the necessity of shifts and the non-extendability to bounded or nested configurations, along with corollaries yielding sumsets or under certain density constraints. The work thereby strengthens the density version of Hindman’s theorem and provides a robust dynamical framework for density-analog sumset patterns with unbounded summand counts. The methods have potential implications for understanding higher-order combinatorial structures in dense sets via infinite-step nilsystems and pronilfactored dynamics.

Abstract

For any set of natural numbers with positive upper Banach density, we show the existence of an infinite set and sequences of natural numbers such that , for every . This strengthens the density finite sums theorem of Kra, Moreira, Richter, and Robertson. We further show, given such a set , the existence of an infinite set and a sequence of natural numbers such that , for every . As a corollary, we obtain a sequence of infinite sets of natural numbers such that , for every . We also establish the optimality of our main theorems by providing counterexamples to potential further generalizations, and thereby addressing questions of the aforementioned authors in the context of density analogs to Hindman's finite sums theorem.
Paper Structure (20 sections, 33 theorems, 157 equations, 2 figures)

This paper contains 20 sections, 33 theorems, 157 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Dynamical reformulation and key ingredients
  4. Notation
  5. Negative results and optimality of the main theorems
  6. Dynamical interpretation
  7. Classical terminology
  8. Translation to dynamics
  9. Progressive measures and infinite-step nilsystems
  10. Pronilfactors
  11. Progressive measures
  12. \ref{['topological pronilfactors 2']} via progressive measures.
  13. The measure $\sigma$ is progressive
  14. Proof of \ref{['topological pronilfactors 2']}
  15. \ref{['topological pronilfactors 1']} via multiple right-progressive measures
  16. ...and 5 more sections

Key Result

Theorem 1

Let $A\subset\mathbb{N}$ be a set with $\mathop{}\!\mathrm{d}^*(A)>0$ and $\ell\geq 0$ an integer. Then, there exist an infinite set $B_{\ell} \subset \mathbb{N}$ and a sequence $(t_{\ell,k})_{k\in \mathbb{N}}\subset \mathbb{N}$, such that

Figures (2)

  • Figure 1: Diagram of the proof of \ref{['topological pronilfactors 2']}
  • Figure 2: Diagram of the proof of \ref{['topological pronilfactors 1']}

Theorems & Definitions (73)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.2
  • Remark 2.3
  • proof : Proof of \ref{['1.2 is optimal (ii)']}
  • Proposition 2.4
  • Remark
  • ...and 63 more