Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the structure of zero-sum free set with minimum subset sums in abelian groups
76
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $G$ be an additive abelian group and $Ssubset G$ 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 $0 otin Sigma(S)$. It was conjectured by R.B.~Eggleton and P.~Erd{o}s in 1972 and proved by W.~Gao et. al. in 2008 that $|Sigma(S)|geq 19$ 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 $|Sigma(S)|=19$.
rate research
Read More
The purpose of the article is to provide an unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let $B(G)$ denote the set consisting of all nonempty zero-sum sequences over G. For $Omega subset B(G$), let $d_{Omega}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|geq t$ has a subsequence in $Omega$.We provide some first results and open problems on $d_{Omega}(G)$.
In 1990, Alon and Kleitman proposed an argument for the sum-free subset problem: every set of n nonzero elements of a finite Abelian group contains a sum-free subset A of size |A|>frac{2}{7}n. In this note, we show that the argument confused two different randomness. It applies only to the finite Abelian group G = (Z/pZ)^s where p is a prime. For the general case, the problem remains open.
A subset $D$ of an Abelian group is $decomposable$ if $emptyset e Dsubset D+D$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $D$ of an Abelian group contains a non-empty subset $Zsubset D$ with $sum Z=0$. For every $ninmathbb N$ we present a decomposable subset $D$ of cardinality $|D|=n$ in the cyclic group of order $2^n-1$ such that $sum D=0$, but $sum T e 0$ for any proper non-empty subset $Tsubset D$. On the other hand, we prove that every decomposable subset $Dsubsetmathbb R$ of cardinality $|D|le 7$ contains a non-empty subset $Zsubset D$ of cardinality $|Z|lefrac12|D|$ with $sum Z=0$. For every $ninmathbb N$ we present a subset $Dsubsetmathbb Z$ of cardinality $|D|=2n$ such that $sum Z=0$ for some subset $Zsubset D$ of cardinality $|Z|=n$ and $sum T e 0$ for any non-empty subset $Tsubset D$ of cardinality $|T|<n=frac12|D|$. Also we prove that every finite decomposable subset $D$ of an Abelian group contains two non-empty subsets $A,B$ such that $sum A+sum B=0$.
We show that, in contrast to the integers setting, almost all even order abelian groups $G$ have exponentially fewer maximal sum-free sets than $2^{mu(G)/2}$, where $mu(G)$ denotes the size of a largest sum-free set in $G$. This confirms a conjecture of Balogh, Liu, Sharifzadeh and Treglown.
Let $vec{w} = (w_1,dots, w_n) in mathbb{R}^{n}$. We show that for any $n^{-2}leepsilonle 1$, if [#{vec{xi} in {0,1}^{n}: langle vec{xi}, vec{w} rangle = tau} ge 2^{-epsilon n}cdot 2^{n}] for some $tau in mathbb{R}$, then [#{langle vec{xi}, vec{w} rangle : vec{xi} in {0,1}^{n}} le 2^{O(sqrt{epsilon}n)}.] This exponentially improves the $epsilon$ dependence in a recent result of Nederlof, Pawlewicz, Swennenhuis, and Wk{e}grzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
