The Density Finite Sums Theorem

Abstract

For any set $A$ of natural numbers with positive upper Banach density and any $k\ge 1$, we show the existence of an infinite set $B\subset \mathbb{N}$ and a shift $t\ge 0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$ for all $m\in \{1,\dots ,k\}$. This can be viewed as a density analog of Hindman's finite sums theorem. Our proof reveals the natural relationships among infinite sumsets, the dynamics underpinning arithmetic progressions, and homogeneous spaces of nilpotent Lie groups.

28 pages. Improved exposition in response to referee's comments