The Cauchy-davenport Theorem for Semigroups

We generalize the Davenport transform to prove that, for A = (A,+) a cancellative unital semigroup and X,Y subsets of A such that the smallest subsemigroup generated by Y is commutative, one has that |X + Y | ≥ Ω(X,Y ) := min ( |X|+ |Y | − 1, sup y0∈Y× min y∈Y \{y0} ord(y − y0) ) if 2 ≤ |X|, |Y | < ∞. While extending the Cauchy-Davenport theorem to the broader and abstract setting of (possibly non-commutative) semigroups, this also strengthens a previous generalization by G. Károlyi relating to sum-sets in commutative groups, where the right-hand side in the above estimate is actually replaced with min(p(A), |X|+ |Y | − 1), with p(A) being the order of the smallest non-trivial subgroup of A. In fact, we show that this bound does not make better than Ω(X,Y ), and is actually (much) lower in some significant situations. Moreover, we prove that the result implies, as a rather immediate corollary, an extension of I. Chowla’s generalization of the Cauchy-Davenport theorem to arbitrary moduli.