Related papers: A New Proof of Kemperman's Theorem

A well-known result by Kemperman describes the structure of those pairs (A,B) of finite subsets of an abelian group satisfying |A+B|\le|A|+|B|-1. We establish a description which is, in a sense, dual to Kemperman's, and as an application…

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…

Let $G\cong \mathbb Z/m_1\mathbb Z\times\ldots\times \mathbb Z/m_r\mathbb Z$ be a finite abelian group with $m_1\mid\ldots\mid m_r=\exp(G)$. The Kemperman Structure Theorem characterizes all subsets $A,\,B\subseteq G$ satisfying…

We obtain critical pair theorems for subsets S and T of an abelian group such that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rodseth and one of the authors.

A classical result of Kemperman gives a complete recursive description of the structure of those subsets $A$ and $B$ of an abelian group that fail to satisfy the triangle inequality, i.e., $|A+B|<|A|+|B|$. In this paper, we achieve the…

Martin Kneser proved the following addition theorem for every abelian group $G$. If $A,B \subseteq G$ are finite and nonempty, then $|A+B| \ge |A+K| + |B+K| - |K|$ where $K = \{g \in G \mid g+A+B = A+B \}$. Here we give a short proof of…

We introduce the notion of a hyper-atom and prove a basic property of this object. This new method allows to improve several results in the classical critical pair theory including its cornerstone: the Kemperman Structure Theorem.

We introduce the notion of a hyper-atom. One of the main results of this paper is the $\frac{2|G|}3$--Theorem: Let $S$ be a finite generating subset of an abelian group $G$ of order $\ge 2$. Let $T$ be a finite subset of $G$ such that $2\le…

In the present work, we introduce the notion of a hyper-atom and prove their main structure theorem. We then apply the global isoperimetric methodology to give a new proof for Kemperman's structure Theory and a slight improvement.

Let $G$ be a $\sigma$-finite abelian group, i.e. $G=\bigcup_{n\geq 1} G_n$ where $(G_n)_{n\geq 1}$ is a non decreasing sequence of finite subgroups. For any $A\subset G$, let $\underline{\mathrm{d}}(A):=\liminf_{n\to\infty}\frac{|A\cap…

We classify the pairs of subsets (A,B) of a locally compact abelian group satisfying m(A+B)=m(A)+m(B), where m is Haar measure. This generalizes a result of M. Kneser classifying such pairs under the additional assumption that G is compact…

Let A and B be subsets of Z/pZ such that |A+B| < |A|+|B|+2. We prove that, if |A|>3, |B|>4, |A+B|<p-4 and p > 52, then A and B are included in arithmetic progressions with the same difference and of size |A|+2 and |B|+2 respectively. This…

Let $G = (G,+)$ be a compact connected abelian group, and let $\mu_G$ denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath and Raikov) establishes the bound $$ \mu_G(A + B) \geq \min(…

For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A =…

Let G be an additive, finite abelian group. The critical number $\mathsf{cr}(G)$ of $G$ is the smallest positive integer $\ell$ such that for every subset $S \subset G \setminus \{0\}$ with $|S| \ge \ell$ the following holds: Every element…

We show in this paper that a sub-critical pair $(A,B)$ of sufficiently "spread-out" Borel sets in a compact and second countable group $K$ with an \emph{abelian} identity component, must reduce to a Sturmian pair in either $\bT$ or $\bT…

This paper establishes a classification of the critical numbers for restricted sumsets in finite abelian groups, determining them exactly for even-order groups and bounding them for odd-order groups, while revealing a fundamental structural…

We give a new equivalent restatement and a new proof in terms of trios to the classical Kneser's theorem. In the finite case, our restatement takes the following, particularly symmetric shape: if $A$, $B$, and $C$ are subsets of a finite…

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is…

Let $G$ be a torsion-free, finitely-generated, nilpotent and metabelian group. In this work we show that $G$ embeds into the group of orientation preserving $C^{1+\alpha}$-diffeomorphisms of the compact interval, for all $\alpha< 1/k$ where…