Related papers: A Structure Theory for Small Sum Subsets

We study pairs of subsets $A, B$ of a compact abelian group $G$ where the sumset $A+B:=\{a+b: a\in A, b\in B\}$ is small. Let $m$ and $m_{*}$ be Haar measure and inner Haar measure on $G$, respectively. Given $\varepsilon>0$, we classify…

Let $G$ be an additive finite abelian group of order $n$, and let $S$ be a sequence of $n+k$ elements in $G$, where $k\geq 1$. Suppose that $S$ contains $t$ distinct elements. Let $\sum_n(S)$ denote the set that consists of all elements in…

A famous result of Freiman describes the structure of finite sets A of integers with small doubling property. If |A + A| <= K|A| then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here…

Given a finite abelian group $G$ and a subset $S\subseteq G$, we let $N_{G,\ S}$ be the smallest integer $N$ such that for any subset $A\subseteq G$ with $N$ elements, we have $g+S\subseteq A$ for some $g\in G$. Using the probabilistic…

These are extended notes of a course given at Tulane University for the 2015 Clifford Lectures. Their aim is to present structure results for group schemes of finite type over a field, with applications to Picard varieties and automorphism…

We develop structure theory of finite Lie conformal superalgebras.

We show, under some natural conditions, that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $\dim X$ in a…

Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in…

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…

We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the paper. These results have applications in various…

Suppose that G is an abelian group, A is a finite subset of G with |A+A|< K|A| and eta in (0,1] is a parameter. Our main result is that there is a set L such that |A cap Span(L)| > K^{-O_eta(1)}|A| and |L| = O(K^eta log |A|). We include an…

Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$ A\stackrel{S}+B:=\{a+b:\,a\in A,\ b\in B,\ a-b\not\in S\} $$ contains at least $$ \min\{|A|+|B|-3|S|,p(G)\} $$…

The article presents several methods for the arithmetic of finite abelian groups. We introduce a tool - already used by Delsarte in [1] as I found out later - analogous to Dirichlet's convolution to obtain combinatorial results on these…

Let $G$ be a finite abelian group and $S$ a sequence with elements of $G$. Let $|S|$ denote the length of $S$ and $\mathrm{supp}(S)$ the set of all the distinct terms in $S$. For an integer $k$ with $k\in [1, |S|]$, let $\Sigma_{k}(S)…

In this article we study the algebraic structure of fine Mordell--Weil groups, plus/minus Mordell--Weil groups, Selmer groups, and plus/minus Selmer groups in the cyclotomic $\mathbb{Z}_p$-extensions of abelian number fields. As a first, we…

We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$ where…

Previously the second author has constructed by cobordism methods, an invariant associated to a finite group $G$. This invariant approximates the number of subgroups of a group, giving in some cases the number of abelian and cyclic…

We present a new structure theorem for finite fields of odd order that relates multiplicative and additive structure in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev…

Let $G$ be an additive abelian group and $S\subset G$ a subset. Let $\Sigma(S)$ denote the set of group elements which can be expressed as a sum of a nonempty subset of $S$. We say $S$ is zero-sum free if $0 \not\in \Sigma(S)$. It was…

A group is metabelian if its commutator subgroup is abelian. For finitely generated metabelian groups, classical commutative algebra, algebraic geometry and geometric group theory, especially the latter two subjects, can be brought to bear…