This article demonstrates infinite linear configurations in sets of positive density, suggesting broader implications for understanding patterns.
In this article we describe all possible infinite linear configurations that can be found in a shift of any set of positive upper Banach density. This simultaneously generalizes Szemer\'edi's theorem on arithmetic progressions and the recent density finite sums theorem of Kra, Moreira, Richter, and Robertson.
No takes yet. Share an insight, caveat, or question.
Felipe Hernández (2025) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: