Sumsets contained in sets of upper Banach density 1

TL;DR

The paper proves that sets of positive integers with upper Banach density 1 contain infinite disjoint subsets of the same density, whose finite sumsets are also contained within the original set, revealing rich additive structure.

Contribution

It establishes the existence of infinite disjoint dense subsets within sets of density 1 with their finite sumsets also contained in the original set, a new structural insight.

Findings

Existence of infinite disjoint subsets with density 1 within sets of density 1.

Finite sumsets of these subsets are contained in the original set.

Reveals complex additive structure in high-density sets.

Abstract

Every set $A$ of positive integers with upper Banach density 1 contains an infinite sequence of pairwise disjoint subsets $(B_i)_{i=1}^{\infty}$ such that $B_i$ has upper Banach density 1 for all $i \in \mathbf{N}$ and $\sum_{i\in I} B_i \subseteq A$ for every nonempty finite set $I$ of positive integers.