The Cauchy-Davenport theorem for semigroups
Résumé
We generalize the Davenport transform to prove that, for $\mathfrak A = (A, +)$ a cancellative unital semigroup and $X,Y$ subsets of $\mathfrak A$ such that $\langle Y \rangle_\mathfrak{A}$ is commutative, one has \begin{displaymath} \textstyle |X + Y| \ge \min \! \big( |X| + |Y| - 1, \sup_{y_0 \in Y^{\times}} \min_{y \in Y \setminus \{y_0\}} {\rm ord}_\mathfrak{A}(y - y_0)\big) \end{displaymath} if $2 \le |X|,|Y| < \infty$. 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 each $\omega(X,Y)$ in the above estimate is replaced with the order of the smallest non-trivial subgroup of $\mathfrak A$, which is never greater than $\omega(X,Y)$, and indeed (much) smaller in some significant situations. Moreover, we show that the result includes, as a special case, I. Chowla's generalization of the Cauchy-Daveport theorem to arbitrary cyclic groups.
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