Cauchy-Davenport type inequalities, I

Abstract

Let $\mathbb{G}=(G,+)$ be a group (either abelian or not). Given $X,Y\subseteq G$, we denote by $⟨Y⟩$ the subsemigroup of $\mathbb{G}$ generated by $Y$, and we set $$\gamma (Y):=\underset{y_0\in Y}{\sup }\underset{y_0\ne y\in Y}{\inf }\mathrm{ord}(y-y_0)$$ if $|Y|\ge 2$ and $\gamma (Y):=|Y|$ otherwise. We prove that if $⟨Y⟩$ is commutative, $Y$ is non-empty, and $X+2Y\ne X+Y+y$ for some $y\in Y$, then $$|X+Y|\ge |X|+\min (\gamma (Y),|Y|-1).$$ Actually, this is obtained from a more general result, which improves on previous work of the author on sumsets in cancellative semigroups, and yields a comprehensive generalization, and in some cases a considerable strengthening, of various additive theorems, notably including the Chowla-Pillai theorem (on sumsets in finite cyclic groups) and the specialization to abelian groups of the Hamidoune-Shatrowsky theorem.

12 pages, no figures. Fixed a mistake from the previous version and, in so doing, obtained a somewhat better inequality (Theorem 2). The paper is a sequel of arXiv:1210.4203 and arXiv:1307.8396 (in particular, it improves on, and subsumes, all the results from the former)