The Structure of Critical Product Sets

Abstract

Let $G$ be a multiplicative group, let $A,B\subseteq G$ be finite and nonempty, and define the product set $AB = {ab \mid $a \in A$and$b \in B$}$. Two fundamental problems in combinatorial number theory are to find lower bounds on $|AB|$, and then to determine structural properties of $A$ and $B$ under the assumption that $|AB|$ is small. We focus on the extreme case when $|AB|<|A|+|B|$, and call any such pair $(A,B)$ \emph{critical}. In the case when $|G|$ is prime, the Cauchy-Davenport Theorem asserts that $|AB|\ge \min |G|,|A|+|B|-1$, and Vosper refined this result by classifying all critical pairs in these groups. For abelian groups, Kneser proved a natural generalization of Cauchy-Davenport by showing that there exists $H\le G$ so that $|AB|\ge |A|+|B|-|H|$ and $ABH=AB$. Kemperman then proved a result which characterizes the structure of all critical pairs in abelian groups. Our main result gives a classification of all critical pairs in an arbitrary group $G$. As a consequence of this we derive the following generalization of Kneser's Theorem to arbitrary groups: There exists $H\le G$ so that $|AB|\ge |A|+|B|-|H|$ and so that for every $y\in AB$ there exists $x\in G$ so that $y(x^{-1}Hx)\subseteq AB$.

147 pages, 28 figures

References in corpus (4)

• Hyper-atoms and the Kemperman's critical pair Theory
• A New Proof of Kemperman's Theorem
• On Minkowski product size: The Vosper's property
• A nilpotent Freiman dimension lemma

Cited by in corpus (3)

• Minimal and nearly minimal measure expansions in connected unimodular groups
• A Cauchy-Davenport theorem for locally compact groups
• An inverse theorem for an inequality of Kneser