The Density Finite Sums Theorem

2025/04/08 by Bryna Kra, Joel Moreira, Kra, Bryna +5 · 2 citations
Computer Science · Mathematics · #37A05 37A44 37B20 05D10 #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2504.06424

openalex publication_date 2025/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any set A of natural numbers with positive upper Banach density and any k≥ 1, we show the existence of an infinite set B⊂\mathbb N and a shift t≥0 such that A-t contains all sums of m distinct elements from B for all m∈\1,…,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.

Cited by

• Adherence Semigroups and Density Finite-Sums Configurations
• On density analogs of Hindman's finite sums theorem

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• Abundant configurations in sumsets with one dense summand
• Infinite linear patterns in sets of positive density
• Infinite unrestricted sumsets of the form B+B in sets with large density
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