The Freiman--Ruzsa Theorem over Finite Fields

Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman--Ruzsa theorem asserts that if |A+A| < K|A| then A is contained in a coset of a subgroup of G of size at most r^{K^4}K^2|A|. It was conjectured by Ruzsa that the subgroup size can be reduced to r^{CK}|A| for some absolute constant C >= 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in fact, yielded a tight bound. In this work, we establish the same conjecture for any prime torsion

Direct and inverse problems for restricted signed sumsets in integers

Let $A=\{a_0, a_1,\ldots, a_{k-1}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$ $(\leq k)$, we let $h^{\wedge}_{\pm}A = \{\Sigma_{i=0}^{k-1} \lambda_{i} a_{i}: \lambda_{i} \in \{-1,0,1\} \text{ for } i=0, 1, \ldots, k-1,~~\Sigma_{i=0}^{k-1} |\lambda_{i}|=h\},$ be the $h$-fold restricted signed sumset of $A$. The direct problem&nbsp;for the restricted signed sumset is to find the minimum number of elements in $h^{\wedge}_{\pm}A$ in terms of $\lvert A\rvert$, where $\lvert A\rvert$ is the cardinality of $A$. The {\it inverse problem} for the restricted signed sumset is to determine the structure of the finite set $A$ for which the minimum value of $|h^{\wedge}_{\pm}A|$ is achieved. In this article, we solve some cases of both direct and inverse problems for $h^{\wedge}_{\pm}A$ in the group of integers. In this connection, we also mention some conjectures in the remaining cases

Direct and inverse problems for restricted signed sumsets in integers

Let $A=\{a_0, a_1,\ldots, a_{k-1}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$ $(\leq k)$, we let $h^{\wedge}_{\pm}A = \{\Sigma_{i=0}^{k-1} \lambda_{i} a_{i}: \lambda_{i} \in \{-1,0,1\} \text{ for } i=0, 1, \ldots, k-1,~~\Sigma_{i=0}^{k-1} |\lambda_{i}|=h\},$ be the $h$-fold restricted signed sumset of $A$. The direct problem&nbsp;for the restricted signed sumset is to find the minimum number of elements in $h^{\wedge}_{\pm}A$ in terms of $\lvert A\rvert$, where $\lvert A\rvert$ is the cardinality of $A$. The {\it inverse problem} for the restricted signed sumset is to determine the structure of the finite set $A$ for which the minimum value of $|h^{\wedge}_{\pm}A|$ is achieved. In this article, we solve some cases of both direct and inverse problems for $h^{\wedge}_{\pm}A$ in the group of integers. In this connection, we also mention some conjectures in the remaining cases

Sumsets with a minimum number of distinct terms

For a non-empty $k$-element set $A$ of an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote this set as $h^{(\geq r)}A$ for integers $h \geq r$. The set $h^{(\geq r)}A$ generalizes the classical sumsets $hA$ and $h\hat{}A$ for $r=1$ and $r=h$, respectively. Thus, we call the set $h^{(\geq r)}A$ the generalized sumset of $A$. By writing the sumset $h^{(\geq r)}A$ in terms of the sumsets $hA$ and $h\hat{}A$, we obtain the sharp lower bound on the size of $h^{(\geq r)}A$ over the groups $\mathbb{Z}$ and $\mathbb{Z}_p$, where $p$ is a prime number. We also characterize the set $A$ for which the lower bound on the size of $h^{(\geq r)}A$ is tight in these groups. Further, using some elementary arguments, we prove an upper bound for the minimum size of $h^{(\geq r)}A$ over the group $\mathbb{Z}_m$ for any integer $m \geq 2$.Comment: 16 page

Some new inequalities in additive combinatorics

In the paper we find new inequalities involving the intersections $A\cap (A-x)$ of shifts of some subset $A$ from an abelian group. We apply the inequalities to obtain new upper bounds for the additive energy of multiplicative subgroups and convex sets and also a series another results on the connection of the additive energy and so--called higher moments of convolutions. Besides we prove new theorems on multiplicative subgroups concerning lower bounds for its doubling constants, sharp lower bound for the cardinality of sumset of a multiplicative subgroup and its subprogression and another results.Comment: 39 page