Abundant configurations in sumsets with one dense summand

TL;DR

This paper demonstrates that sumsets with one infinite set and a dense summand contain rich arithmetic structures, including expected densities of progressions and specific finite configurations, contrasting prior limitations.

Contribution

It proves that sumsets with an infinite set and a positive density set contain expected densities of progressions and finite configurations, expanding understanding of their combinatorial structure.

Findings

Sumsets contain at least the expected density of k-term arithmetic progressions.

Sumsets must contain finite configurations not necessarily in arbitrary positive density sets.

Contrasts with previous results limiting configurations in sets of positive density.

Abstract

We analyze sumsets A+B = {a+b : a in A, b in B} where A,B are sets of integers, A is infinite, and B has positive upper Banach density. For each k, we show that A+B contains at least the expected density of k-term arithmetic progressions based on the density of B, in contrast with an example of Bergelson, Host, Kra, and Ruzsa. Furthermore, we show that A+B must contain finite configurations not necessarily found in arbitrary sets of positive density, in contrast with results of Frantzikinakis, Lesigne, and Wierdl on sets of k-recurrence.