A Cauchy-Davenport theorem for semigroups

Abstract

We generalize the Davenport transform and use it to prove that, for a (possibly non-commutative) cancellative semigroup $\mathbb{A}=(A,+)$ and non-empty subsets $X,Y$ of $A$ such that the subsemigroup generated by $Y$ is commutative, we have $|X+Y|\ge \min (\omega (Y),|X|+|Y|-1)$, where $\omega (Y):={\sup }_{y_0\in Y\cap {\mathbb{A}}^{\times }}{\inf }_{y\in Y\setminus \{y_0\}}|<y-y_0>|$. This carries over the Cauchy-Davenport theorem to the broader setting of semigroups, and it implies, in particular, an extension of I. Chowla's and S.S. Pillai's theorems for cyclic groups and a notable strengthening of another generalization of the same Cauchy-Davenport theorem to commutative groups, where $\omega (Y)$ in the above is replaced by the minimal order of the non-trivial subgroups of $\mathbb{A}$.

14 pages, to appear in Uniform Distribution Theory. Fixed minor details w.r.t. the previous version

References in corpus (1)

• Small doubling in ordered semigroups

Cited by in corpus (2)

• Cauchy-Davenport type theorems for semigroups
• Cauchy-Davenport type inequalities, I