Related papers: Signed Difference Sets

A symmetric subset of the reals is one that remains invariant under some reflection z --> c-z. We consider, for any 0 < x <= 1, the largest real number D(x) such that every subset of $[0,1]$ with measure greater than x contains a symmetric…

In this paper we study higher-order difference equations which can be written as follows: $$ \mathbf{J} (y_0,y_1,...)^T = \lambda^N (y_0,y_1,...)^T, $$ where $\mathbf{J}$ is a $(2N+1)$-diagonal bounded banded matrix…

The distinguishing number $D(G,X)$ of an action of a group $G$ on a set $X$ is the least size of a partition of $X$ such that no element of $G$ acting nontrivially on $X$ preserves this partition. In this paper we describe the…

Let $G$ be a group and let $V$ be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts $\Sigma \subset V^G$ which generalize both the class of algebraic sofic subshifts of $V^G$ and the class of…

Let $G$ be a graph. For a subset $X$ of $V(G)$, the switching $\sigma$ of $G$ is the signed graph $G^{\sigma}$ obtained from $G$ by reversing the signs of all edges between $X$ and $V(G)\setminus X$. Let $A(G^{\sigma})$ be the adjacency…

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

An achievement set of a series is a set of all its subsums. We study the properties of achievement sets of conditionally convergent series in finite dimensional spaces. The purpose of the paper is to answer some of the open problems…

Let $A=\{a_1,a_2,\dots, a_m\}$ be a subset of a finite abelian group $G$. We call $A$ {\it $t$-independent} in $G$, if whenever $$\lambda_1a_1+\lambda_2a_2+\cdots +\lambda_m a_m=0$$ for some integers $\lambda_1, \lambda_2, \dots ,…

In this work we prove that the set of the difference of primes is a $\Delta_r^*$-set. The work is based on the recent dramatic new developments in the study of bounded gaps between primes, reached by Zhang, Maynard and Tao.

The $k$-dominating graph $D_k(G)$ of a graph $G$ is defined on the vertex set consisting of dominating sets of $G$ with cardinality at most $k$, two such sets being adjacent if they differ by either adding or deleting a single vertex. A…

Let $G$ be a linear algebraic group acting linearly on a vector space $V$, and let $k[V]^G$ be the corresponding algebra of invariant polynomial functions. A separating set $S \subseteq k[V]^G$ is a set of polynomials with the property that…

A signed graph is a graph with a function that assigns a label of positive or negative to each edge. The sign of a circle is the product of the signs of its edges; a graph is balanced if all of its circles are positive. A set of edges whose…

Let $G$ be a finite group. The ring $R_\mathbb{K}(G)$ of virtual characters of $G$ over the field $\mathbb{K}$ is a $\lambda$-ring; as such, it is equipped with the so-called $\Gamma$-filtration, first defined by Grothendieck. We explore…

Let $v$ be a positive odd integer. A $(v,k,\lambda)$-perfect difference family (PDF) is a collection $\mathcal{F}$ of $k$-subsets of $\{0,1,\ldots,v-1\}$ such that the multiset $\bigcup_{F\in \mathcal{F}}\{x-y : x,y\in F, x>y\}$ covers each…

Consider a group $G$ acting on a set $\Omega$, the vector $v_{a,b}$ is a vector with the entries indexed by the elements of $G$, and the $g$-entry is 1 if $g$ maps $a$ to $b$, and zero otherwise. A $(G,\Omega)$-Cameron-Liebler set is a…

Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of $]n[:= \{1,2,..., n\}$ such that elements of A are incongruent modulo p and non-zero modulo p. Let $k \geq…

Given a real number $\lambda > 1$, we say that $d|n$ is a $\lambda$-middle divisor of $n$ if $$ \sqrt{\frac{n}{\lambda}} < d \leq \sqrt{\lambda n}. $$ We will prove that there are integers having an arbitrarily large number of…

Let $G$ be a group. A subset $D$ of $G$ is a determining set of $G$, if every automorphism of $G$ is uniquely determined by its action on $D$. The determining number of $G$, denoted by $\alpha(G)$, is the cardinality of a smallest…

We present a difference analogue of a result given by Hrushovski on differential Galois groups under specialization. Let $k$ be an algebraically closed field of characteristic zero and $\mathbb{X}$ an irreducible affine algebraic variety…

We define a special sort of weighted oriented graphs, signed quivers. Each of these yields a symmetric quiver, i.e., a quiver endowed with an involutive anti-automorphism and the inherited signs. We develop a representation theory of…