On density analogs of Hindman's finite sums theorem

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.