Multiplicative structures in additively large sets

Previous research extending over a few decades has established that multiplicatively large sets (in any of several interpretations) must have substantial additive structure. We investigate here the question of how much multiplicative structure can be found in additively large sets. For example, we show that any translate of a set of finite sums from an infinite sequence must contain all of the initial products from another infinite sequence. And, as a corollary of a result of Renling Jin, we show that if A and B have positive upper Banach density, then A + B contains all of the initial products from an infinite sequence. We also show that if a set has a complement which is not additively piecewise syndetic, then any translate of that set is both additively and multiplicatively large in several senses. We investigate whether a subset of N with bounded gaps – a syndetic set – must contain arbitrarily long geometric progressions. We believe that we establish that this is a significant open question.