Related papers: Sets Arising as Minimal Additive Complements in th…

For a fixed $c > 0$ we construct an arbitrarily large set $B$ of size $n$ such that its sum set $B+B$ contains a convex sequence of size $cn^2$, answering a question of Hegarty.

Given a finite group $G$ and positive integers $r$ and $s$, a problem of interest in algebra is determining the minimum cardinality of the product set $AB$, where $A$ and $B$ are subsets of $G$ such that $|A|=r$ and $|B|=s$. This problem…

In this article, we study connections between components of the Cayley graph $\mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and cosets of the subgroup of $G$ generated by $A$. In particular, we show how to construct…

Given an Abelian groups $G$, denote $\mu(G)$ the size of its largest sum-free subset and $f_{\max}(G)$ the number of maximal sum-free sets in $G$. Confirming a prediction by Liu and Sharifzadeh, we prove that all even-order $G\ne…

This article introduces descriptive fixed sets and their properties in descriptive proximity spaces viewed in the context of planar ribbon complexes. These fixed sets are a byproduct of descriptive proximally continuous maps that spawn…

Let $\mathcal{E}=\text{Rep}(G)$ be a Tannakian fusion category. For a braided fusion category $\mathcal{C}$ over $\mathcal{E}$ we give sufficient and necessary conditions that characterize the Witt relation $[\mathcal{C}]=[\mathcal{E}]$.…

We prove that the complement of any non-separating planar graph of order $2n-3$ contains a $K_n$ minor, and argue that the order $2n-3$ is lowest possible with this property. To illustrate the necessity of the non-separating hypothesis, we…

Our focus is on the set of lower-triangular, infinite matrices that have natural operations like addition, multiplication by a number, and matrix multiplication. With respect to addition this set forms and abelian group while with respect…

There is a long history of studying Ramsey theory using the algebraic structure of the Stone-\v{C}ech compactification of discrete semigroup. It has been shown that various Ramsey theoretic structures are contained in different algebraic…

For an abelian topological group G let G^* denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G) and an open…

Given an algebra $F[H]^G$ of polynomial invariants of an action of the group $G$ over the vector space $H$, a subset $S$ of $F[H]^G$ is called separating if $S$ separates all orbits that can be separated by $F[H]^G$. A minimal separating…

Let $G$ be a multiplicative group, let $A,B \subseteq G$ be finite and nonempty, and define the product set $AB = {ab \mid $a \in A$ and $b \in B$}$. Two fundamental problems in combinatorial number theory are to find lower bounds on…

Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see \cite{...} for introduction. In \cite{shokurov:hyp} it was conjectured that many of the interesting sets, associated with these…

We present sufficient conditions for the existence of matchings in abelian groups and their linear counterparts. These conditions lead to extensions of existing results in matching theory. Additionally, we classify subsets within abelian…

We prove that if $d \ge 2$ is an integer, $G$ is a finite abelian group, $Z_0$ is a subset of $G$ not contained in any strict coset in $G$, and $E_1,\dots,E_d$ are dense subsets of $G^n$ such that the sumset $E_1+\dots+E_d$ avoids $Z_0^n$…

Two observables are called complementary if preparing a physical object in an eigenstate of one of them yields a completely random result in a measurement of the other. We investigate small sets of complementary observables that cannot be…

For a finite abelian group $G$, let $\beta_{\mathrm{sep}}(G)$ denote its separating Noether number. We determine $\beta_{\mathrm{sep}}(G)$ exactly for every finite abelian group $ G \cong C_{n_1}\oplus \cdots \oplus C_{n_r}$ with $ 1<n_1…

The complement of a complex hyperplane arrangement is known to be homotopic to a minimal CW complex. There are several approaches to the minimality. In this paper, we restrict our attention to real two dimensional cases, and introduce the…

Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote the minimum…

If $\mathscr{G} = (G, +)$ is an abelian group, $S \subset G$ is said to scatter under addition if for all $a,b \in S$, $a+b \not \in S$. If $\mathscr{U}^{n}_{p}$ is the set of $n$th roots of unity in $\mathbb{Z}/p\mathbb{Z}$, where $n \geq…