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Bela Bajnok

Publications and source records attributed to Bela Bajnok.

At least 19 recordsLinked to original sources
arXiv2024

The spanning number and the independence number of a subset of an abelian group

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 $$λ_1a_1+λ_2a_2+\cdots +λ_m a_m=0$$ for some integers $λ_1, λ_2, \dots , λ_m$ with $$|λ_1|+|λ_2|+\cdots +|λ_m| \leq t,$$ we have $λ_1=λ_2= \cdots = λ_m=0$, and we say that $A$ is {\it $s$-spanning} in $G$, if every element $g$ of $G$ can be written as $$g=λ_1a_1+λ_2a_2+\cdots +λ_m a_m$$ for some integers $λ_1, λ_2, \dots , λ_m$ with $$|λ_1|+|λ_2|+\cdots +|λ_m| \leq s.$$ In this paper we give an upper bound for the size of a $t$-independent set and a lower bound for the size of an $s$-spanning set in $G$, and determine some cases when this extremal size occurs. We also discuss an interesting connection to spherical combinatorics.

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arXiv2024

Orbits of the hyperoctahedral group as Euclidean designs

The hyperoctahedral group $H$ in $n$ dimensions (the Weyl group of Lie type $B_n$) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes. A finite set ${\cal X} \subset \mathbb{R}^n$ with a weight function $w: {\cal X} \rightarrow \mathbb{R}^+$ is called a Euclidean $t$-design, if $$\sum_{r \in R} W_r \overline{f}_{S_{r}} = \sum_{{\bf x} \in {\cal X}} w({\bf x}) f({\bf x})$$ holds for every polynomial $f$ of total degree at most $t$; here $R$ is the set of norms of the points in ${\cal X}$, $W_r$ is the total weight of all elements of ${\cal X}$ with norm $r$, $S_r$ is the $n$-dimensional sphere of radius $r$ centered at the origin, and $\overline{f}_{S_{r}}$ is the average of $f$ over $S_{r}$. Here we consider Euclidean designs which are supported by orbits of the hyperoctahedral group. Namely, we prove that any Euclidean design on a union of generalized hyperoctahedra has strength (maximum $t$ for which it is a Euclidean design) equal to 3, 5, or 7. We find explicit necessary and sufficient conditions for when this strength is 5 and for when it is 7. In order to establish our classification, we translate the above definition of Euclidean designs to a single equation for $t=5$, a set of three equations for $t=7$, and a set of seven equations for $t=9$. Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), proved a Fisher-type inequality $|{\cal X}| \geq N(n,p,t)$ for the minimum size of a Euclidean $t$-design in $\mathbb{R}^n$ on $p=|R|$ concentric spheres (assuming that the design is antipodal if $t$ is odd). A Euclidean design with exactly $N(n,p,t)$ points is called tight. We exhibit new examples of antipodal tight Euclidean designs, supported by orbits of the hyperoctahedral group, for $N(n,p,t)=$(3,2,5), (3,3,7), and (4,2,7).

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arXiv2024

On the Diameter of Undirected Cayley Graphs of Finite Abelian Groups

Let $s$ be a positive integer. Our goal is to find all finite abelian groups $G$ that contain a $2$-subset $A$ for which the undirected Cayley graph $Γ(G,A)$ has diameter at most $s$. We provide a complete answer when $G$ is cyclic, and a conjecture and some partial answers when $G$ is noncyclic.

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arXiv2024

Constructions of Spherical 3-Designs

Spherical t-designs are Chebyshev-type averaging sets on the d-sphere S^d which are exact for polynomials of degree at most t. This concept was introduced in 1977 by Delsarte, Goethals, and Seidel, who also found the minimum possible size of such designs, in particular, that the number of points in a 3-design on S^d must be at least n>=2d+2. In this paper we give explicit constructions for spherical 3-designs on S^d consisting of n points for d=1 and n>=4; d=2 and n=6; 8; >= 10; d=3 and n=8; >=10; d = 4 and n = 10; 12; >= 14; d>=5 and n>=5(d+1)/2 odd or n>=2d+2 even. We also provide some evidence that 3-designs of other sizes do not exist.

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arXiv2024

An historical overview of the influence of technology on mathematical competitions

We provide an historical overview of how advances in technology influenced high school and university mathematical competitions in the United States and at the International Mathematical Olympiad. While students are not allowed the usage of technological aids during mathematical competitions, the developments in technology (especially graphing technology) throughout the past century and the increasing employment of such aids in the classroom have affected both the nature of the proposed problems and their expected solutions. We examine several interesting examples from competitions going back several decades.

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arXiv2022

On Perfect Bases in Finite Abelian Groups

Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of $A$; similarly, we say that $A$ is a {\em perfect restricted $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ distinct elements of $A$. We prove that perfect $s$-bases exist only in the trivial cases of $s=1$ or $|A|=1$. The situation is different with restricted addition where perfection is more frequent; here we treat the case of $s=2$ and prove that $G$ has a perfect restricted $2$-basis if, and only if, it is isomorphic to $\mathbb{Z}_2$, $\mathbb{Z}_4$, $\mathbb{Z}_7$, $\mathbb{Z}_2^2$, $\mathbb{Z}_2^4$, or $\mathbb{Z}_2^2 \times \mathbb{Z}_4$.

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arXiv2022

On sumsets of nonbases of maximum size

Let $G$ be a finite abelian group. A nonempty subset $A$ in $G$ is called a basis of order $h$ if $hA=G$; when $hA \neq G$, it is called a nonbasis of order $h$. Our interest is in all possible sizes of $hA$ when $A$ is a nonbasis of order $h$ in $G$ of maximum size; we provide the complete answer when $h=2$ or $h=3$.

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arXiv2022

Additive Combinatorics in Groups and Geometric Combinatorics on Spheres

We embark on a tour that takes us through four closely related topics: the dual concepts of independence and spanning in finite abelian groups and the analogous dual concepts of designs and distance sets on spheres. We review some of the main known results in each area, mention several open questions, and discuss some connections among these four interesting topics.

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arXiv2017

Additive Combinatorics: A Menu of Research Problems

This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for student research, it is hopefully also useful for a wider audience. It is the author's intention to keep the material current, thus all feedback and updates are greatly appreciated.

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arXiv2016

More on the $h$-critical numbers of finite abelian groups

For a finite abelian group $G$, a nonempty subset $A$ of $G$, and a positive integer $h$, we let $hA$ denote the $h$-fold sumset of $A$; that is, $hA$ is the collection of sums of $h$ not-necessarily-distinct elements of $A$. Furthermore, for a positive integer $s$, we set $[0,s] A=\cup_{h=0}^s h A$. We say that $A$ is a generating set of $G$ if there is a positive integer $s$ for which $[0,s] A=G$. The $h$-critical number $χ(G,h)$ of $G$ is defined as the smallest positive integer $m$ for which $hA=G$ holds for every $m$-subset $A$ of $G$; similarly, $χ(G,[0,s])$ is the smallest positive integer $m$ for which $[0,s]A=G$ holds for every $m$-subset $A$ of $G$. We define $\widehatχ (G, h)$ as the smallest positive integer $m$ for which $hA=G$ holds for every generating $m$-subset $A$ of $G$; $\widehatχ (G, [0,s])$ is defined similarly. The value of $χ(G,h)$ has been determined by this author for all $G$ and $h$, and $\widehatχ (G, [0,s])$ was introduced and resolved for some special cases by Klopsch and Lev. Here we determine the remaining two quantities in all cases.

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