Related papers: On Freiman's 3k-4 theorem

The $3k-4$ conjecture in groups $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime states that if $A$ is a nonempty subset of $\mathbb{Z}/p\mathbb{Z}$ satisfying $2A\neq \mathbb{Z}/p\mathbb{Z}$ and $|2A|=2|A|+r \leq \min\{3|A|-4,\;p-r-4\}$, then $A$ is…

Motivated by the definition of Freiman homomorphism, we explore the possibilities of formulating some basic notions and techniques of additive combinatorics in a categorical language. We show that additive sets and Freiman homomorphisms…

The $3k-4$ Theorem asserts that, if $A,\,B\subseteq \mathbb Z$ are finite, nonempty subsets with $|A|\geq |B|$ and $|A+B|=|A|+|B|+r< |A|+2|B|-3$, then there are arithmetic progressions $P_A$ and $P_B$ of common difference with $X\subseteq…

We take the first step toward a structure theory that includes both operations of a ring $\mathcal{R}$. More precisely, we prove a series of inverse results for the structure of sets $A\subseteq \mathbf{F}_p$ such that, under certain…

We determine the structure of a finite subset $A$ of an abelian group given that $|2A|<3(1-\epsilon)|A|$, $\epsilon>0$; namely, we show that $A$ is contained either in a "small" one-dimensional coset progression, or in a union of fewer than…

Let $\mathbb{F}_q$ be a finite field of order $q$. Iosevich and Rudnev (2005) proved that for any set $A\subset \mathbb{F}_q^d$, if $|A|\gg q^{\frac{d+1}{2}}$, then the distance set $\Delta(A)$ contains a positive proportion of all…

The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent…

We show that a finite set of integers $A \subseteq \mathbb{Z}$ with $|A+A| \le K |A|$ contains a large piece $X \subseteq A$ with Fre\u{i}man dimension $O(\log K)$, where large means $|A|/|X| \ll \exp(O(\log^2 K))$. This can be thought of…

The theory of finitely supported algebraic structures is related to Pitts theory of nominal sets (by equipping finitely supported sets with finitely supported internal algebraic laws). It represents a reformulation of Zermelo Fraenkel set…

We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

It has been proved in \cite{ge} for every $p$-group of order $p^n$, $|\mathcal{M}(G)|=p^{\f{1}{2}n(n-1)-t(G)}$, where $t(G)\geq 0$. In \cite{be, el, zh}, the structure of $G$ has been characterized for $t(G)=0,1,2,3$ by several authors.…

The well--known Freiman--Ruzsa Theorem provides a structural description of a set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$--dimensional arithmetic progression $P$ with $|P|\le c'|A|$, where $d$ and $c'$ depend only on $c$.…

The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements…

A set system is called union closed if for any two sets in the set system their union is also in the set system. Gilmer recently proved that in any union closed set system some element belongs to at least a $0.01$ fraction of sets, and…

Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some…

Let G be any additive abelian group with cyclic torsion subgroup, and let A, B and C be finite subsets of G with cardinality n>0. We show that there is a numbering {a_i}_{i=1}^n of the elements of A, a numbering {b_i}_{i=1}^n of the…

We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put forward an algebraic interpretation of the…

Freiman's theorem asserts, roughly speaking, if that a finite set in a torsion-free abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by…

The usual definition of the set of constructible reals is $\Sigma ^1_2$. This set can have a simpler definition if, for example, it is countable or if every real is constructible. H. Friedman asked if the set of constructible reals can be…