Remarks on a generalization of the Davenport constant

A generalization of the Davenport constant is investigated. For a finite abelian group G and a positive integer k, let denote the smallest ? such that each sequence over G of length at least ? has k disjoint non-empty zero-sum subsequences. For general G, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence is eventually an arithmetic progression with difference exp(G), and several questions arising from this fact are investigated. For elementary 2-groups, is investigated in detail; in particular, the exact values are determined for groups of rank four and five (for rank at most three they were already known).     

Davenport constant with weights and some related questions, II

Let G be a finite abelian group of order n and let A⊆Z be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by DA(G), to be the least natural number k such that for any sequence (x₁,…,xk) with xi∈G, there exists a non-empty subsequence (xj₁,…,xjl) and a₁,…,al∈A such that . Similarly, for any such set A, EA(G) is defined to be the least t∈N such that for all sequences (x₁,…,xt) with xi∈G, there exist indices j₁,…,jn∈N,1?j₁<?<jn?t, and ?₁,…,?n∈A with . In the present paper, we establish a relation between the constants DA(G) and EA(G) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group G=Z/nZ and the relation we establish had been conjectured in that particular case.     

Correction to: Davenport constant for semigroups

Semigroup Forum - Given a finite commutative semigroup (written additively), denote by     

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

We classify real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. It follows from this classification that all of them are open parts of homogeneous ones.

     

Davenport constant for finite abelian groups

For a finite abelian group G, we investigate the length of a sequence of elements of G that is guaranteed to have a subsequence with product identity of G. In particular, we obtain a bound on the length which takes into account the repetitions of elements of the sequence, the rank and the invariant factors of G. Consequently, we see that there are plenty of such sequences whose length could be much shorter than the best known upper bound for the Davenport constant of G, which is the least integer s such that any sequence of length s in G necessarily contains a subsequence with product identity. We also show that the Davenport constant for the multiplicative group of reduced residue classes modulo n is comparatively large with respect to the order of the group, which is φ(n),when n is in certain thin subsets of positive integers. This is done by studying the Carmichael’s lambda function, defined as the maximal multiplicative order of any reduced residue modulo n, along these subsets.     

Constructing sparse Davenport–Schinzel sequences

For any sequence $u$, the extremal function $Ex(u,j,n)$ is the maximum possible length of a $j$-sparse sequence with $n$ distinct letters that avoids $u$. We prove that if $u$ is an alternating sequence $abab\dots$ of length $s$, then $Ex(u,j,n)=\Theta \left(s{n}^2\right)$ for all $j\ge 2$ and $s\ge n$, answering a question of Wellman and Pettie (2018) and extending the result of Roselle and Stanton that $Ex(u,2,n)=\Theta \left(s{n}^2\right)$ for any alternation $u$ of length $s\ge n$ (Roselle and Stanton, 1971).Wellman and Pettie also asked how large must $s\left(n\right)$ be for there to exist $n$-block $DS(n,s\left(n\right))$ sequences of length $\Omega \left(n^{2-o\left(1\right)}\right)$. We answer this question by showing that the maximum possible length of an $n$-block $DS(n,s\left(n\right))$ sequence is $\Omega \left(n^{2-o\left(1\right)}\right)$ if and only if $s\left(n\right)=\Omega \left(n^{1-o\left(1\right)}\right)$. We also show related results for extremal functions of forbidden 0–1 matrices with any constant number of rows and extremal functions of forbidden sequences with any constant number of distinct letters.     

Lower bounds on Davenport–Schinzel sequences via rectangular Zarankiewicz matrices

An order-$s$Davenport–Schinzel sequence over an $n$-letter alphabet is one avoiding immediate repetitions and alternating subsequences with length $s+2$. The main problem is to determine the maximum length of such a sequence, as a function of $n$ and $s$. When $s$ is fixed this problem has been settled (see Agarwal, Sharir, and Shor, 1989, Nivasch, 2010 and Pettie, 2015) but when $s$ is a function of $n$, very little is known about the extremal function $\lambda (s,n)$ of such sequences.In this paper we give a new recursive construction of Davenport–Schinzel sequences that is based on dense 0–1matrices avoiding large all-1 submatrices (aka Zarankiewicz’s Problem ). In particular, we give a simple construction of $n^{2∕t}\times n$ matrices containing $n^{1+1∕t}$ 1s that avoid $t\times 2$ all-1 submatrices. (This result seems to be absent from the literature on Zarankiewicz’s problem, but it may be considered folklore among experts in this area [Z. Füredi, personal communication, 2017].)Our lower bounds on $\lambda (s,n)$ exhibit three qualitatively different behaviors depending on the size of $s$ relative to $n$. When $s\le loglogn$ we show that $\lambda (s,n)∕n\ge 2^s$ grows exponentially with $s$. When $s=n^{o\left(1\right)}$ we show $\lambda (s,n)∕n\ge {\left(\frac{s}{2log{log}_{s}n}\right)}^{log{log}_{s}n}$ grows faster than any polynomial in $s$. Finally, when $s=\Omega (n^{1∕t}(t?1)!)$, $\lambda (s,n)=\Omega (n^{2}s∕(t?1)!)$ matches the trivial upper bound $O\left(n^{2}s\right)$ asymptotically, whenever $t$ is constant.     

On three approaches to conjugacy in semigroups

We compare three approaches to the notion of conjugacy for semigroups, the first one via the transitive closure of the uv∼vu relation, the second one via an action of inverse semigroups on themselves by partial transformations, and the third one via characters of finite-dimensional representations.     

Generation of the integrated semigroups by superelliptic differential operators

Let A be a superelliptic differential operator of order 2m introduced by E.B. Davies [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169]. In the case of 2m>N, he obtained the upper Gaussian bound of the integral kernel representing (e−zA)_z∈C+ and the estimates of the Lp-operator norm of the semigroup for all p∈[1,∞). The purpose of the present paper is to show that −i(A+k) (for some constant k>0) generates an integrated semigroup on Lα,p (weighted Lp space) and lp(Lα,q). To prove this we need norm estimates of (e−zA)_z∈C+ on each of these spaces. Also we get another norm estimate of (e−zA)_z∈C+ on Lp when 2m>N without using the integral kernel. This norm estimate is better than that in [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169] and gives a better “times of the integration” of the integrated semigroup.     

Digital representation of semigroups and groups

A digital representation of a semigroup (S,⋅) is a family 〈F _t 〉 _t∈I , where I is a linearly ordered set, each F _t is a finite non-empty subset of S and every element of S is uniquely representable in the form Π _t∈H   x _t where H is a finite subset of I, each x _t ∈F _t and products are taken in increasing order of indices. (If S has an identity 1, then Π _t∈∅   x _t =1.) A strong digital representation of a group G is a digital representation of G with the additional property that for each t∈I, for some x _t ∈G and some m _t >1 in ℕ where m _t =2 if the order of x _t is infinite, while, if the order of x _t is finite, then m _t is a prime and the order of x _t is a power of m _t . We show that any free semigroup has a digital representation with each | F _t |=1 and that each Abelian group has a strong digital representation. We investigate the problem of whether all groups, or even all finite groups have strong digital representations, obtaining several partial results. Finally, we give applications to the algebra of the Stone-Čech compactification of a discrete group and the weakly almost periodic compactification of a discrete semigroup. Dedicated to Karl Heinrich Hofmann on the occasion of his 75th birthday. Stefano Ferri was partially supported by a research grant of the Faculty of Sciences of Universidad de los Andes. The support is gratefully acknowledged. Neil Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803.     

Extension operators via semigroups

The Roper-Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is A-spirallike for a variety of linear operators A. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.     

Quasi-hyperbolic semigroups

We introduce the notion of quasi-hyperbolic operators and C₀-semigroups. Examples include the push-forward operator associated with a quasi-Anosov diffeomorphism or flow. A quasi-hyperbolic operator can be characterised by a simple spectral property or as the restriction of a hyperbolic operator to an invariant subspace. There is a corresponding spectral property for the generator of a C₀-semigroup, and it characterises quasi-hyperbolicity on Hilbert spaces but not on other Banach spaces. We exhibit some weaker properties which are implied by the spectral property.     

Extremal product-one free sequences in Dihedral and Dicyclic Groups

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D\left(G\right)$ such that every sequence of $G$ with $D\left(G\right)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be the Dihedral Group of order $2n$ and $Q_{4n}$ be the Dicyclic Group of order $4n$. Zhuang and Gao (2005) showed that $D\left(D_{2n}\right)=n+1$ and Bass (2007) showed that $D\left(Q_{4n}\right)=2n+1$. In this paper, we give explicit characterizations of all sequences $S$ of $G$ such that $\left|S\right|=D\left(G\right)?1$ and $S$ is free of subsequences whose product is 1, where $G$ is equal to $D_{2n}$ or $Q_{4n}$ for some $n$.