Related papers: Semicontinuity of structure for small sumsets in c…

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…

We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and…

Following Darji, we say that a Borel subset $B$ of an abelian Polish group $G$ is Haar meager if there is a compact metric space $K$ and a continuous function $f : K \to G$ such that the preimage of the translate, $f^{-1}(B+g)$ is meager in…

Let A and B be subsets of an elementary abelian 2-group G, none of which are contained in a coset of a proper subgroup. Extending onto potentially distinct summands a result of Hennecart and Plagne, we show that if |A+B|<|A|+|B|, then…

Let $G$ be a locally compact abelian group with Haar measure $\mu$. For integers $n \geq 2$ and $H \geq 2$ and for any $n$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_H \in \mathbf{N}^n$, there exist measurable subsets $A_1,\ldots, A_n$ of $G$…

Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such…

Given a locally compact group bundle, we show that the system of the Plancherel weights of their C*-algebras is lower semi-continuous. As a corollary, we obtain that the dual Haar sytem of a continuous Haar system of a locally compact…

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(…

We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in…

We develop a new method leading the structure of finite subsets S and T of an abelian group with $|S+T|\le |S|+|T|$. We show also how to recover the known results in this area in a relatively short space.

If $A$ and $B$ are subsets of an abelian group, their sumset is $A+B:=\{a+b:a\in A, b\in B\}$. We study sumsets in discrete abelian groups, where at least one summand has positive upper Banach density. Renling Jin proved that if $A$ and $B$…

In this paper we construct a compact quantum semigroup structure on the Toeplitz algebra $\mathcal{T}$. The existence of a subalgebra, isomorphic to the algebra of regular Borel's measures on a circle with convolution product, in the dual…

Let $G$ be a compact abelian group and $\phi_1, \phi_2, \phi_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $\phi_i$'s, we prove the existence of Bohr sets in the sumset $\phi_1(A) + \phi_2(A) + \phi_3(A)$,…

Let $G$ be a connected unimodular group equipped with a (left and hence right) Haar measure $\mu_G$, and suppose $A, B \subseteq G$ are nonempty and compact. An inequality by Kemperman gives us…

Let $A_1$ and $A_2$ be randomly chosen subsets of the first $n$ integers of cardinalities $s_2\geq s_1 = \Omega(s_2)$, such that their sumset $A_1+A_2$ has size $m$. We show that asymptotically almost surely $A_1$ and $A_2$ are almost fully…

Suppose $G$ is a connected noncompact locally compact group, $A,B$ are nonempty and compact subsets of $G$, $\mu$ is a left Haar measure on $G$. Assuming that $G$ is unimodular, and $ \mu(A^2) < K \mu(A) $ with $K>1$ a fixed constant, our…

In this paper we are interested in the following notions of smallness: a subset $A$ of an abelian Polish group $X$ is called Haar-countable/Haar-finite/Haar-$n$ if there are a Borel hull $B\supseteq A$ and a copy $C$ of $2^\omega$ such that…

Inspired by recent questions of Nathanson, we show that for any infinite abelian group $G$ and any integers $m_1, \ldots, m_H$, there exist finite subsets $A,B \subseteq G$ such that $|hA|-|hB|=m_h$ for each $1 \leq h \leq H$. We also…

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…

In this paper, we describe the relationship between the quasi-component q(G) of a (perfectly) minimal pseudocompact abelian group G and the quasi-component q(\widetilde G) of its completion. Specifically, we characterize the pairs (C,A) of…