On density analogs of Hindman's finite sums theorem

TL;DR

This paper extends Hindman's finite sums theorem to sets of natural numbers with positive upper Banach density, demonstrating the existence of structured sumsets within such dense sets and establishing optimality through counterexamples.

Contribution

The paper introduces new density analogs of Hindman's theorem, showing structured sumsets exist in dense sets and proving the results are optimal with counterexamples.

Findings

Existence of infinite sets B and sequences (t_k), (s_k) with sumsets contained in A-t_k.

Existence of infinite sets B with sumsets of fixed size contained in A-t_k.

Construction of sequences (B_n) with sumsets contained in A for all n.

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.