Let $A$ be a subset of the cyclic group $\mathbf{Z}/p\mathbf{Z}$ with $p$ prime. It is a well-studied problem to determine how small $|A|$ can be if there is no unique sum in $A+A$, meaning that for every two elements $a_1,a_2\in A$, there exist $a_1',a_2'\in A$ such that $a_1+a_2=a_1'+a_2'$ and $\{a_1,a_2\}\neq \{a_1',a_2'\}$. Let $m(p)$ be the size of a smallest subset of $\mathbf{Z}/p\mathbf{Z}$ with no unique sum. The previous best known bounds are $\log p \ll m(p)\ll \sqrt{p}$. In this paper we improve both the upper and lower bounds to $ω(p)\log p \leqslant m(p)\ll (\log p)^2$ for some function $ω(p)$ which tends to infinity as $p\to \infty$. In particular, this shows that for any $B\subset \mathbf{Z}/p\mathbf{Z}$ of size $|B|<ω(p)\log p$, its sumset $B+B$ contains a unique sum. We also obtain corresponding bounds on the size of the smallest subset of a general Abelian group having no unique sum.
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 $$ \min\{|A|+|B|-3|S|,p(G)\} $$ elements, where $p(G)$ is the least prime factor of $|G|$. Further, we also have $$ |A\stackrel{S}+B|\geq \min\{|A|+|B|-|S|-2,p(G)\}, $$ provided that both $|A|$ and $|B|$ are large with respect to $|S|$.
In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we present some results about the validity of these conjectures.
For a finite abelian group $G$ with subsets $A$ and $B$, the sumset $AB$ is $\{ab \mid a\in A, b \in B\}$. A fundamental problem in additive combinatorics is to find a lower bound for the cardinality of $AB$ in terms of the cardinalities of $A$ and $B$. This article addresses the analogous problem for sequences in abelian groups and flags in field extensions. For a positive integer $n$, let $[n]$ denote the set $\{0,\dots,n-1\}$. To a finite abelian group $G$ of cardinality $n$ and an ordering $G = \{1=v_0,\dots,v_{n-1}\}$, associate the function $T \colon [n] \times [n] \rightarrow [n]$ defin…
We show that a zero-sum-free sequence of length $n$ over an abelian group spans at least $2n$ distinct subsequence sums, unless it possesses a rigid, easily-described structure.
Let $G$ be an additive abelian group and $S\subset G$ a subset. Let $Σ(S)$ 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 $0 \not\in Σ(S)$. It was conjectured by R.B.~Eggleton and P.~Erdös in 1972 and proved by W.~Gao et. al. in 2008 that $|Σ(S)|\geq 19$ provided that $S$ is a zero-sum free subset of an abelian group $G$ with $|S|=6$. In this paper, we determined the structure of zero-sum free set $S$ where $|S|=6$ and $|Σ(S)|=19$.
Motivated by a problem in additive Ramsey theory, we extend Todorcevic's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for every Abelian group $G$ of size $\aleph_2$, there exists a coloring $c:G\rightarrow\mathbb Z$ such that for every uncountable $X\subseteq G$ and every integer $k$, there are three distinct elements $x,y,z$ of $X$ such that $c(x+y+z)=k$.
We show that a finite zero-sum-free sequence $α$ over an abelian group has at least $c|α|^{4/3}$ distinct subsequence sums, unless $α$ is "controlled" by a small number of its terms; here $|α|$ denotes the number of terms of $α$, and $c>0$ is an absolute constant.
Let $A$ be a multiset with elements in an abelian group. Let $FS(A)$ be the multiset containing the $2^{|A|}$ sums of all subsets of $A$. We study the reconstruction problem ``Given $FS(A)$, is it possible to identify $A$?'', and we give a satisfactory answer for all abelian groups. We prove that, up to identifying multisets through a natural equivalence relation, the function $A \mapsto FS(A)$ is injective (and thus the reconstruction problem is solvable) if and only if every order $n$ of a torsion element of the abelian group satisfies a certain number-theoretical property linked to the mult…
We address the "sums of dilates problem" by looking for non-trivial lower bounds on sumsets of the form $k \cdot X + l \cdot X$, where $k$ and $l$ are non-zero integers and $X$ is a subset of a possibly non-abelian group $G$ (written additively). In particular, we investigate the extension of some results so far known only for the integers to the context of torsion-free or linearly orderable groups, either abelian or not.
Flows that are sums of hamiltonian cycles in Cayley graphs on abelian groups
如果X是任意有限阿贝尔群上的连通Cayley图，我们精确地确定了X上的哪些流可以写成哈密顿圈的和。(This Brian Alspach的回答）。特别地，如果X的次数至少为5，并且X具有偶数个顶点，则可以这样写的流恰好是偶数流，即流f，使得f的边流之和可被2整除。另一方面，也有4次的例子，其中不是所有的偶流都可以写成哈密顿圈的和。类似的结果是已知的，从工作的阿尔斯帕奇，洛克和维特，为情况下，X是立方的，或有一个奇数的顶点。
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 all pairs $A,B$ of Haar measurable subsets of $G$ satisfying $m(A), m(B)>\varepsilon$ and $m_{*}(A+B)\leq m(A)+m(B)+δ$ where $δ=δ(\varepsilon)>0$ is small. We also study the case where the $δ$-popular sumset $A+_δB:=\{t\in G: m(A\cap (t-B))>δ\}$ is small. We prove that for all $\varepsilon>0$, there is a $δ>0$ such that if $A$ and $B$ are subsets of a compact …
A $Γ$-magic rectangle set $MRS_Γ(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $Γ$, each appearing once, with all row sums in every rectangle equal to a constant $ω\in Γ$ and all column sums in every rectangle equal to a constant $δ\in Γ$. In this paper we prove that for $\{a,b\}\neq\{2^α,2k+1\}$ where $α$ and $k$ are some natural numbers, a $Γ$-magic rectangle set MRS$_Γ(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|Γ|$ is odd or $Γ$ has more than one involution. Moreover we obtain sufficient and necessary conditio…
Let $A$ be a set in an abelian group $G$. For integers $h,r \geq 1$ the generalized $h$-fold sumset, denoted by $h^{(r)}A$, is the set of sums of $h$ elements of $A$, where each element appears in the sum at most $r$ times. If $G=\mathbb{Z}$ lower bounds for $|h^{(r)}A|$ are known, as well as the structure of the sets of integers for which $|h^{(r)}A|$ is minimal. In this paper we generalize this result by giving a lower bound for $|h^{(r)}A|$ when $G=\mathbb{Z}/p\mathbb{Z}$ for a prime $p$, and show new proofs for the direct and inverse problems in $\mathbb{Z}$.
Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible cardinality of the difference set $A-A$, which is always greater than or equal to the smallest possible cardinality of $A+A$ and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that $G$ is a cyclic group or a vector space over a finite field. This resolves a conjecture of Bajnok and Matzke on signed sum…
A subset of an abelian group is {\em sequenceable} if there is an ordering $(x_1, \ldots, x_k)$ of its elements such that the partial sums $(y_0, y_1, \ldots, y_k)$, given by $y_0 = 0$ and $y_i = \sum_{j=1}^i x_i$ for $1 \leq i \leq k$, are distinct, with the possible exception that we may have $y_k = y_0 = 0$. We demonstrate the sequenceability of subsets of size $k$ of $\mathbb{Z}_n \setminus \{ 0 \}$ when $n = mt$ in many cases, including when $m$ is either prime or has all prime factors larger than $k! /2$ for $k \leq 11$ and $t \leq 5$ and for $k=12$ and $t \leq 4$. We obtain similar, but…
Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group $G$ is called a CI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $S^α=T$ for some automorphism $α$ of $G$. A finite group $G$ is called a BI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $M_ν^S=M_ν^T$ for all positive integers $ν$, where $M_ν…
Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions
Faculty of Engineering, Özyĕğin University（Özyeğin大学工程学院）；Department of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences（西挪威应用科学大学计算机科学、电气工程和数学科学系）；Applied Mathematics Department, Naval Postgraduate School（海军研究生院应用数学系）
When Do Restricted Sumsets Cover a Finite Abelian Group?
School of Mathematics, Southwest Jiaotong University（西南交通大学数学学院）；School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）
Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z\rbrace$ such that $x+y=z$. We show that $f(n,2) = 2^{\lceil n/2\rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r \leq 5$. Similar results were obtained by Hàn and Jiménez in the setting of finite abelian groups.
