Cauchy-Davenport Theorem for Semigroups

We generalize the Davenport transform and use it to prove that, for a (possibly non-commutative) cancellative semigroup A = (A, +) and non- empty subsets X, Y of A such that the subsemigroup generated by Y is commu- tative, we have |X + Y | ≥ min(ω(Y ), |X| + |Y | − 1), where ω(Y ) := sup inf |y − y 0|. y0∈Y ∩A× y∈Y \{y0} This carries over the Cauchy-Davenport theorem to the broader setting of semi- groups, and it implies, on the one hand, a common extension of I. Chowla’s and S.S. Pillai’s theorems for cyclic groups, and on the other a significant strength- ening of another generalization of the same Cauchy-Davenport theorem (to com- mutative groups), where ω(Y ) in the above is replaced by the infimum of |S| as S ranges over the non-trivial subgroups of A.