The Density Finite Sums Theorem

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.