Abundant configurations in sumsets with one dense summand

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.