Notions of Size and Combinatorial Properties of Quotient Sets in Semigroups

An IP* set in a semigroup is one which must intersect the set of finite products from any specified sequence. (If the semigroup is noncommutative, one must specify the order of the products, resulting in “left” and “right” IP* sets.) If A is a subset of N with positive upper density, then the difference set A−A = {x ∈ N : there exists y ∈ A with x+ y ∈ A} is an IP* set in (N,+). Defining analogously the quotient sets AA−1 and A−1A, we analyze notions of largeness sufficient to guarantee that one or the other of these quotient sets are IP* sets. Among these notions are thick , syndetic, and piecewise syndetic sets, all of which come in both “left” and “right” versions. For example, we show that if A is any left syndetic subset of a semigroup S, then AA−1 is both a left IP* set and a right IP* set, while A−1A need be neither a left IP* set nor a right IP* set, even in a group. We also investigate the relationships among these notions of largeness.