New 2--critical sets in the abelian 2--group

Polynomial Criterion for Abelian Difference Sets

Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function.

Spherical Designs and Generalized Sum-Free Sets in Abelian Groups

We extend the concepts of sum-free sets and Sidon-sets of combinatorial number theory with the aim to provide explicit constructions for spherical designs. We call a subset $S$ of the (additive) abelian group $G$ {\it $t$-free} if for all non-negative integers $k$ and $l$ with $k+l \leq t$, the sum of $k$ (not necessarily distinct) elements of $S$ does not equal the sum of $l$ (not necessarily distinct) elements of $S$ unless $k=l$ and the two sums contain the same terms. Here we shall give asymptotic bounds for the size of a largest $t$-free set in ${\bf Z}_n$, and for $t \leq 3$ discuss how …

A characterisation of elementary abelian 3-groups

Tarnauceanu [Archiv der Mathematik, 102 (1), (2014), 11--14] gave a characterisation of elementary abelian $2$-groups in terms of their maximal sum-free sets. His theorem states that a finite group $G$ is an elementary abelian $2$-group if and only if the set of maximal sum-free sets coincides with the set of complements of the maximal subgroups. A corollary is that the number of maximal sum-free sets in an elementary abelian $2$-group of finite rank $n$ is $2^n-1$. Regretfully, we show here that the theorem is wrong. We then prove a correct version of the theorem from which the desired coroll…

Sets with few differences in abelian groups

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…

Critical sets in the elementary abelian 2- and 3- groups

In 1998, Khodkar showed that the minimal critical set in the Latin square corresponding to the elementary abelian 2-group of order 16 is of size at most 124. Since the paper was published, improved methods for solving integer programming problems have been developed. Here we give an example of a critical set of size 121 in this Latin square, found through such methods. We also give a new upper bound on the size of critical sets of minimal size for the elementary abelian 2-group of order $2^n$: $4^{n}-3^{n}+4-2^{n}-2^{n-2}$. We speculate about possible lower bounds for this value, given some ot…

Semi-regular Relative Difference Sets with Large Forbidden Subgroups

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that there does not exist a $(2p,p,2p,2)$ relative difference set in any group of order $2p^2$, and an abelian $(4p,p,4p,4)$ relative difference set can only exist in the group $\Bbb{Z}_2^2\times \Bbb{Z}_3^2$. On the other hand, we construct a family of non-abelian relative difference sets with parameters $(4q,q,4q,4)$, where $q$ is an odd prime power greater th…

Negative Latin square type partial difference sets in nonelementary abelian 2-groups

Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, we construct the first known family of partial difference sets with negative Latin square type parameters in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even. Similarly, we construct partial difference sets with Latin square type parameters in the same groups for all $k$ when $\ell$ is even and for all $k<\ell$ when $\ell$ is odd. These constructions provide the first examp…

Counting MSTD Sets in Finite Abelian Groups

In an abelian group G, a more sums than differences (MSTD) set is a subset A of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets in finite abelian groups, extending previous results of Nathanson. The proof contains an application of a recently resolved conjecture of Alon and Kahn on the number of independent sets in a regular graph.

On abelian $(2^{2m+1}(2^{m-1}+1), 2^m(2^m+1), 2^m)$-difference sets

In this paper we prove that an abelian group contains $(2^{2m+1}(2^{m-1}+1), 2^m(2^m+1), 2^m)$-difference sets with $m\geqslant 3$ if and only if it contains an elementary abelian 2-group of order $2^{2m}$. Our proof shows that the method of constructing such difference sets is essentially unique.

虚拟阿贝尔群代数集的多变量增长级数的有理性

Rationality of the multivariate growth series for algebraic sets of virtually abelian groups

本文证明了虚拟阿贝尔群代数集的多变量相对增长级数具有有理性，证实了 Evetts 和 Levine 的猜想。

Non-existence of two types of partial difference sets

In this note we prove the non-existence of two types of partial difference sets in Abelian groups of order 216. This finalizes the classification of parameters for which a partial difference set of size at most 100 exists in an Abelian group.

New necessary conditions for (negative) Latin square type partial difference sets in abelian groups

Partial difference sets (for short, PDSs) with parameters ($n^2$, $r(n-ε)$, $εn+r^2-3εr$, $r^2-εr$) are called Latin square type (respectively negative Latin square type) PDSs if $ε=1$ (respectively $ε=-1$). In this paper, we will give restrictions on the parameter $r$ of a (negative) Latin square type partial difference set in an abelian group of non-prime power order. As far as we know no previous general restrictions on $r$ were known. Our restrictions are particularly useful when $a$ is much larger than $b$. As an application, we show that if there exists an abelian negative Latin square t…

有限阿贝尔群上的幻矩形集的存在性

Existence of magic rectangle sets over finite abelian groups

该研究针对有限阿贝尔群上的G-幻矩形集MRS_G(a,b;c)的存在性问题，推导得出其存在的充要条件，证实了Cichacz与Hinc提出的相关猜想。

非初等阿贝尔 2 - 群中的负拉丁方型部分差集

Negative Latin-Square-Type Partial Difference Sets in Non-Elementary Abelian 2-Groups

该研究利用三次分圆类、特征理论等，在多个非初等阿贝尔 2 - 群中构造负拉丁方型部分差集，前四种构造通过特定划分取代三次分圆划分，得到了指定群中首批具所述参数的部分差集。

Results on formally dual sets in finite abelian groups of size 64 obtained from a graph search algorithm

We shortly present two small results regarding the study formal duality in finite abelian groups as introduced by Cohn, Kumar, Reiher and Schürmann. In particular, we give a new example of a formally self dual set in $\mathbb{Z}_2^2\times\mathbb{Z}_4^2$ and computed nonexistence of primitive formally dual sets of size $8$ in $\mathbb{Z}_8^2$.

Minimal inversion complete sets and maximal abelian ideals

Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak b$ a fixed Borel subalgebra, and $W$ the Weyl group of $\mathfrak g$. In this note, we study a relationship between the maximal abelian ideals of $\mathfrak b$ and the minimal inversion complete sets of $W$. The latter have been recently defined by Malvenuto et al. (J. Algebra, 424 (2015), 330-356.)

Enumerating independent sets in Abelian Cayley graphs

We show that any connected Cayley graph $Γ$ on an Abelian group of order $2n$ and degree $\tildeΩ(\log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $Γ$ is bipartite. Our proof is based on Sapozhenko's graph container method and uses the Plünnecke-Rusza-Petridis inequality from additive combinatorics.

Triple arrays from difference sets

This paper addresses the question whether triple arrays can be constructed from Youden squares developed from difference sets. We prove that if the difference set is abelian, then having $-1$ as multiplier is both a necessary and sufficient condition for the construction to work. Using this, we are able to give a new infinite family of triple arrays. We also give an alternative and more direct version of the construction, leaving out the intermediate step via Youden squares. This is used when we analyse the case of non-abelian difference sets, for which we prove a sufficient condition for givi…

论某些非交换群中不存在斜阿达马差集

On the non-existence of skew-Hadamard difference sets in certain non-abelian groups

本文通过有理群代数结构，证明了若幂零群存在斜阿达马差集，则该群必为p-群，首次给出非交换群中斜阿达马差集的一般结构限制。

差结构的转移：一个新的半直积框架

Transfer of difference structures: a new semidirect product framework

本文提出基于半直积的新框架，实现阿贝尔与非阿贝尔群间差结构（差集、近因子分解、外部差族）的转移，解决开放问题，并构造首个无限族非阿贝尔循环外部差族。

关于阿贝尔群中无和集的注记

Notes on sum-free sets in abelian groups

本文通过容器方法，渐近确定了大多数偶数阶阿贝尔群中极大无和子集的数量，并提出了若干开放问题。

Quantitative structure of stable sets in finite abelian groups

We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was recently obtained by the first author in joint work with Conant and Pillay, using model-theoretic techniques. In contrast, the approach in the present paper is highly quantitative and relies on several key ingredients from arithmetic combinatorics.

Counting configuration-free sets in groups

We provide new examples of the asymptotic counting for the number of subsets on groups of given size which are free of certain configurations. These examples include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. As a consequence, random counterparts are presented as well.

Counting sum-free sets in Abelian groups

In this paper we study sum-free sets of order $m$ in finite Abelian groups. We prove a general theorem on 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sum-free sets of order $m$ in Abelian groups $G$ whose order is divisible by a prime $q$ with $q \equiv 2 \pmod 3$, for every $m \ge C(q) \sqrt{n \log n}$, thus extending and refining a theorem of Green and Ruzsa. In particular, we prove that almost all sum-free subsets of size $m…

New constructions of strongly regular Cayley graphs on abelian groups

Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order $9^iv^4$ for any odd positiv…

Structure of large incomplete sets in abelian groups

Let $G$ be a finite abelian group and $A$ be a subset of $G$. We say that $A$ is complete if every element of $G$ can be represented as a sum of different elements of $A$. In this paper, we study the following question: {\it What is the structure of a large incomplete set ?} The typical answer is that such a set is essentially contained in a maximal subgroup. As a by-product, we obtain a new proof for several earlier results.

Packings of partial difference sets

A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial difference sets in abelian groups having identical parameters, the size of the collection being either the maximum possible or one smaller. We unify and extend numerous previous results in a common framework, recognizing that a particular subgroup reveals imp…

On maximal sum-free sets in abelian groups

Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical Society, 2018] recently gave a sharp count on the number of maximal sum-free subsets of $\{1, \dots, n\}$, thereby answering a question of Cameron and Erdős. In contrast, not as much is know about the analogous problem for finite abelian groups. In this paper we give the first sharp results in this direction, determining asymptotically the number of maximal sum-free sets in both the binary and ternary spaces $\mathbb Z^k_2$ and $\mathbb Z^k_3$. We also make progress on a conjecture of Balogh, Liu, Sharifzadeh and Tregl…

Strongly regular Cayley graphs over primary abelian groups of rank 2

Strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2 were studied in [J.A Davis, Partial difference sets in p-groups, Arch.Math.63 (1994) 103-110; K.H Leung, S.L. Ma, Partial difference sets with Paley parameters, Bull. London Math Soc. 27 (1995) 553-564]. It was shown that such graphs exist iff the corresponding group is isomorphic to ${\mathbb Z}_{p^n} \oplus {\mathbb Z}_{p^n}$, where $p$ is an odd prime. In this paper we classify all strongly regular Cayley graphs over this group using Schur rings method. As a consequence we obtain a complete classification of …

Number of $A+B\ne C$ solutions in abelian groups and application to counting independent sets in hypergraphs

The paper deals with a problem of Additive Combinatorics. Let ${\mathbf G}$ be a finite abelian group of order $N$. We prove that the number of subset triples $A,B,C\subset {\mathbf G}$ such that for any $x\in A$, $y\in B$ and $z\in C$ one has $x+y\ne z$ equals $$ 3\cdot 4^N+N3^{N+1} + O((3-c_*)^N) $$ for some absolute constant $c_*>0$. This provides a tight estimate for the number of independent sets in a special 3-uniform linear hypergraph and gives a support for the natural conjecture concerning the maximal possible number of independent sets in such hypergraphs on $n$ vertices.

Hypermaps over non-abelian simple groups and strongly symmetric generating sets

A generating pair $x, y$ for a group $G$ is said to be \textbf{\textit{symmetric}} if there exists an automorphism $φ_{x,y}$ of $G$ inverting both $x$ and $y$, that is, $x^{φ_{x,y}}=x^{-1}$ and $y^{φ_{x,y}}=y^{-1}$. Similarly, a group $G$ is said to be \textbf{\textit{strongly symmetric}} if $G$ can be generated with two elements and if all generating pairs of $G$ are symmetric. In this paper we classify the finite strongly symmetric non-abelian simple groups. Combinatorially, these are the finite non-abelian simple groups $G$ such that every orientably regular hypermap with monodromy group $G…

非不相交强外部差族可以具有任意数量的集合

Non-disjoint strong external difference families can have any number of sets

本文通过将不相交性条件替换为非不相交性，构造出具有任意数量集合的阿贝尔强外部差族，从而绕过了经典SEDF关于多于2个集合的唯一性猜想，并揭示了其与经典SEDF的差异、联系及新的编码应用来源。

Sum-free sets in abelian groups

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a constant depending on G and lying in the interval [2/7,1/2]. We also estimate the number of sum-free subsets of G. It turns out that log_2 of this number is c(G)|G| + o(|G|), which is tight up to the o-term. For certain abelian groups, those whose order is divisible by a small prime of the form 3k + 2, we can obtain an asymptotic for the number of sum-free s…

Maximum number of sum-free colorings in finite abelian groups

An $r$-coloring of a subset $A$ of a finite abelian group $G$ is called sum-free if it does not induce a monochromatic Schur triple, i.e., a triple of elements $a,b,c\in A$ with $a+b=c$. We investigate $κ_{r,G}$, the maximum number of sum-free $r$-colorings admitted by subsets of $G$, and our results show a close relationship between $κ_{r,G}$ and largest sum-free sets of $G$. Given a sufficiently large abelian group $G$ of type $I$, i.e., $|G|$ has a prime divisor $q$ with $q\equiv 2\pmod 3$. For $r=2,3$ we show that a subset $A\subset G$ achieves $κ_{r,G}$ if and only if $A$ is a largest sum…

Arithmetic Sets in Groups

We define a notion of an arithmetic set in an arbitrary countable group and study properties of these sets in the cases of Abelian groups and non-abelian free groups.

Extremal incomplete sets in finite abelian groups

Let $G$ be a finite abelian group. The critical number ${\rm cr}(G)$ of $G$ is the least positive integer $\ell$ such that every subset $A\subseteq G\setminus\{0\}$ of cardinality at least $\ell$ spans $G$, i.e., every element of $G$ can be written as a nonempty sum of distinct elements of $A$. The exact values of the critical number have been completely determined recently for all finite abelian groups. The structure of these sets of cardinality ${\rm cr}(G)-1$ which fail to span $G$ has also been characterized except for the case that $|G|$ is an even number and the case that $|G|=pq$ with $…

Every nonnegative real number is an abelian critical exponent

The abelian critical exponent of an infinite word $w$ is defined as the maximum ratio between the exponent and the period of an abelian power occurring in $w$. It was shown by Fici et al. that the set of finite abelian critical exponents of Sturmian words coincides with the Lagrange spectrum. This spectrum contains every large enough positive real number. We construct words whose abelian critical exponents fill the remaining gaps, that is, we prove that for each nonnegative real number $θ$ there exists an infinite word having abelian critical exponent $θ$. We also extend this result to the $k$…

The largest $(k, \ell)$-sum-free sets in compact abelian groups

A subset $A$ of a finite abelian group is called $(k,\ell)$-sum-free if $kA \cap \ell A=\emptyset.$ In this paper, we extend this concept to compact abelian groups and study the question of how large a measurable $(k,\ell)$-sum-free set can be. For integers $1 \leq k <\ell$ and a compact abelian group $G$, let $$λ_{k,\ell}(G)=\sup\{ μ(A): kA \cap \ell A =\emptyset \}$$ be the maximum possible size of a $(k,\ell)$-sum-free subset of $G$. We prove that if $G=\mathbb{I} \times M$, where $\mathbb{I}$ is the identity component of $G$, then $$λ_{k, \ell}(G)=\max \left\{ λ_{k, \ell}(M), λ_{k, \ell}(\…

组合转移：构造无限族非交换差集、偏差集和相对差集的新方法

Combinatorial transfer: a new method for constructing infinite families of nonabelian difference sets, partial difference sets, and relative difference sets

本文提出组合转移法构造组合对象的非交换自同构群，发现了多个非交换差集、偏差集和相对差集的无限族，并证明了Dillon二面体技巧的逆定理。

A Plünnecke-Ruzsa inequality in compact abelian groups

The Plünnecke-Ruzsa inequality is a fundamental tool to control the growth of finite subsets of abelian groups under repeated addition and subtraction. Other tools to handle sumsets have gained applicability by being extended to more general subsets of more general groups. This motivates extending the Plünnecke-Ruzsa inequality, in particular to measurable subsets of compact abelian groups by replacing the cardinality with the Haar probability measure. This objective is related to the question of the stability of classes of Haar measurable sets under addition. In this direction the class of an…

A note on matchings in abelian groups

The question of finding sets of monomials which are removable from a generic homogeneous polynomial through a linear change in its variables was raised by E. K. Wakeford in 1916. This linear algebra question motivated J. Losonczy to define the concept of acyclic matchings in Z n, and later in abelian groups. In this paper, we give a constructive approach to study the acyclic matchings in cyclic groups. We also introduce the notion of weakly matched subsets and investigate its relation with matchings in abelian groups.

向量空间及其他阿贝尔群中小规模的完全无三项等差数列集合

Complete $3$-term arithmetic progression free sets of small size in vector spaces and other abelian groups

该研究面向阿贝尔群中的无三项等差数列集合，定义并分析 $3$-AP 饱和性，为多类向量空间和循环群构造小规模完全无 $3$-AP 集合及饱和集合。

A new structure for difference matrices over abelian $p$-groups

A difference matrix over a group is a discrete structure that is intimately related to many other combinatorial designs, including mutually orthogonal Latin squares, orthogonal arrays, and transversal designs. Interest in constructing difference matrices over $2$-groups has been renewed by the recent discovery that these matrices can be used to construct large linking systems of difference sets, which in turn provide examples of systems of linked symmetric designs and association schemes. We survey the main constructive and nonexistence results for difference matrices, beginning with a classic…

格雷厄姆关于交换群中小集的猜想

Graham conjecture on small sets in abelian groups

研究了交换群中大小为20以内的子集序列性，证明了其可序列化，改进了零和子集的界限，并探讨了相关CMPP猜想。

Centerpole sets for colorings of Abelian groups

Given a topological group $G$ we calculate or evaluate the cardinal characteristic $c_k(G)$ (and $c_k^B(G)$) equal to the smallest cardinality of a $k$-centerpole subset $C\subset G$ for (Borel) colorings of $G$. A subset $C\subset G$ of a topological group $G$ is called {\em $k$-centerpole} if for each (Borel) $k$-coloring of $G$ there is an unbounded monochromatic subset $G$, which is symmetric with respect to a point $c\in C$ in the sense that $S=cS^{-1}c$.

$S_h$-集在阿贝尔群中

$S_h$-sets in abelian groups

本文研究阿贝尔群中 $S_h$-集的最小群阶下界，建立了 $v_3(k)$ 的新下界，并统一恢复了 $v_2(k)$ 的已知结果。

具有小乘积集的集合的结构，在无 torsion 自由群、素数阶循环群和交换群中

Structure of sets with small product sets in torsion-free groups, cyclic groups of prime orders and abelian groups

本文研究了无 torsion 自由群、素数阶循环群和交换群中具有小乘积集的集合结构，推导了广义乘积集的最优下界，并扩展了子序列和的相关结果。

Sum-full sets are not zero-sum-free

Let $A$ be a finite, nonempty subset of an abelian group. We show that if every element of $A$ is a sum of two other elements, then $A$ has a nonempty zero-sum subset. That is, a (finite, nonempty) sum-full subset of an abelian group is not zero-sum-free.

Zero-sum subsets of decomposable sets in Abelian groups

A subset $D$ of an Abelian group is $decomposable$ if $\emptyset\ne D\subset D+D$. 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 $Z\subset D$ with $\sum Z=0$. For every $n\in\mathbb N$ 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 $\sum T\ne 0$ for any proper non-empty subset $T\subset D$. On the other hand, we prove that every decomposable subset $D\subset\mathbb R$ of cardinality $|D|\le 7$ contains a non-emp…

Abelian networks III. The critical group

The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network. We exhibit the critical group as a quotient of a free abelian group by a subgroup containing the image of the Laplacian, with equality in the case that the network is rectangular. We generalize Dhar's burning algorithm to abelian networks, and estimate the running time of an abelian network on an …

Critical pairs in abelian groups and Kemperman's structure theorem

A well-known result by Kemperman describes the structure of those pairs (A,B) of finite subsets of an abelian group satisfying |A+B|\le|A|+|B|-1. We establish a description which is, in a sense, dual to Kemperman's, and as an application sharpen several results due to Deshouillers, Hamidoune, Hennecart, and Plagne.

Inequalities of Riesz-Sobolev type for compact connected Abelian groups

A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses. A corresponding stability theorem for sets whose sumset has nearly minimal measure is also proved, sharpening recent results of other authors. For the special case of the group $\mathbb{R}/\mathbb{Z}$, a continuous deformation of sets is developed, under which an appropriately scaled Riesz-Sobolev functional is shown to be nondecreasing.

交换群上的交集集

Intersective sets over abelian groups

本文通过代数图论将交换群中的交集集问题与循环多项式联系起来，推广并改进了Alon和Hegedűs的结果，并构造了无限多个非平凡的 $G$ 和 $J$ 家族以指数级改进现有上界。

紧交换群中缺失差集的集合研究

On sets with missing differences in compact abelian groups

本文将Motzkin缺失差问题推广到紧交换群，证明其等价于有限秩商群上的离散问题，并依据格秩判定最大密度避开集是否周期存在，推出至多三个缺失差时密度为有理数。