Cauchy-Davenport type theorems for semigroups

TL;DR

This paper extends the Cauchy-Davenport theorem to non-commutative semigroups, providing new bounds on sumset sizes and generalizing existing inequalities for groups and torsion-free structures.

Contribution

It introduces a novel function (Z) and proves a generalized Cauchy-Davenport inequality for cancellative semigroups, broadening the scope of additive combinatorics.

Findings

Extended Cauchy-Davenport theorem for cancellative semigroups

Generalized Kemperman's inequality for torsion-free groups

Strengthened existing inequalities with new bounds

Abstract

Let $\mathbb{A} = (A, +)$ be a (possibly non-commutative) semigroup. For $Z \subseteq A$ we define $Z^\times := Z \cap \mathbb A^\times$, where $\mathbb A^\times$ is the set of the units of $\mathbb{A}$, and $$\gamma(Z) := \sup_{z_0 \in Z^\times} \inf_{z_0 \ne z \in Z} {\rm ord}(z - z_0).$$ The paper investigates some properties of $\gamma(\cdot)$ and shows the following extension of the Cauchy-Davenport theorem: If $\mathbb A$ is cancellative and $X, Y \subseteq A$, then $$|X+Y| \ge \min(\gamma(X+Y),|X| + |Y| - 1).$$ This implies a generalization of Kemperman's inequality for torsion-free groups and strengthens another extension of the Cauchy-Davenport theorem, where $\mathbb{A}$ is a group and $\gamma(X+Y)$ in the above is replaced by the infimum of $|S|$ as $S$ ranges over the non-trivial subgroups of $\mathbb{A}$ (Hamidoune-K\'arolyi theorem).