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arXiv · 2504.06424

The Density Finite Sums Theorem

Bryna Kra·Joel Moreira·Florian K. Richter·Donald Robertson
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Abstract

For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$ for all $m\in\{1,\ldots,k\}$. 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.

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Bryna Kra, Joel Moreira, Florian K. Richter, Donald Robertson. 2025-04-08. The Density Finite Sums Theorem. https://arxiv.org/abs/2504.06424

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