Davenport Constant Research Articles

Local and Global Tameness in Krull Monoids

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then the global tame degree equals zero if and only if H is factorial (equivalently, |G| = 1). If |G| > 1, then , where is the Davenport constant of G. We analyze the case when equals the lower bound, and we show that grows asymptotically as the upper bound, when both terms are considered as functions of the rank of G. We provide more precise results if G is either cyclic or an elementary 2-group.

Remarks on the plus–minus weighted Davenport constant

For (G, +) a finite abelian group the plus–minus weighted Davenport constant, denoted D±(G), is the smallest ℓ such that each sequence g1…gℓover G has a weighted zero-subsum with weights +1 and -1, i.e. there is a non-empty subset I ⊂ {1,…,ℓ} such that ∑i∈Iaigi= 0 for ai∈ {+1, -1}. We present new bounds for this constant, mainly lower bounds, and also obtain the exact value of this constant for various additional types of groups.

On zero-sum subsequences of length [formula omitted

Let G be an additive finite abelian group of exponent exp(G). For every positive integer k, let skexp(G)(G) denote the smallest integer t such that every sequence over G of length t contains a zero-sum subsequence of length kexp(G). We prove that if exp(G) is sufficiently larger than |G|exp(G) then skexp(G)(G)=kexp(G)+D(G)−1 for all k⩾2, where D(G) is the Davenport constant of G.

An upper bound for the Davenport constant of finite groups

Let G be a finite (not necessarily abelian) group and let p=p(G) be the smallest prime number dividing |G|. We prove that d(G)⩽|G|p+9p2−10p, where d(G) denotes the small Davenport constant of G which is defined as the maximal integer ℓ such that there is a sequence over G of length ℓ containing no nonempty one-product subsequence.

EQUIDISTRIBUTION OF DIVISORS IN RESIDUE CLASSES AND REPRESENTATIONS BY BINARY QUADRATIC FORMS

We study the number of divisors in residue classes modulo m and prove, for example, that the exact equidistribution holds for almost all natural numbers coprime to m in the sense of natural density if and only if m = 2kp1p2…ps, where k and s are non-negative integers and pj are distinct Fermat primes. We also provide a general and exact lower bound for the proportion of divisors in the residue class 1 mod m. The same combinatorial technique using Davenport's constant leads to exact lower bounds for the number of representations of a natural number by a given binary quadratic form with a negative discriminant.

On the generalized Davenport constant and the Noether number

Abstract Known results on the generalized Davenport constant relating zero-sum sequences over a finite abelian group are extended for the generalized Noether number relating rings of polynomial invariants of an arbitrary finite group. An improved general upper degree bound for polynomial invariants of a non-cyclic finite group that cut out the zero vector is given.

Products of two atoms in Krull monoids and arithmetical characterizations of class groups

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and let D(G) be the Davenport constant of G. Then a product of two atoms of H can be written as a product of at most D(G) atoms. We study this extremal case and consider the set U{2,D(G)}(H) defined as the set of all l∈N with the following property: there are two atoms u,v∈H such that uv can be written as a product of l atoms as well as a product of D(G) atoms. If G is cyclic, then U{2,D(G)}(H)={2,D(G)}. If G has rank two, then we show that (apart from some exceptional cases) U{2,D(G)}(H)=[2,D(G)]∖{3}. This result is based on the recent characterization of all minimal zero-sum sequences of maximal length over groups of rank two. As a consequence, we are able to show that the arithmetical factorization properties encoded in the sets of lengths of a rank 2 prime power order group uniquely characterizes the group.

NON-COMMUTATIVE KRULL MONOIDS: A DIVISOR THEORETIC APPROACH AND THEIR ARITHMETIC

A (not necessarily commutative) Krull monoid---as introduced by Wauters---is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these results we give a criterion for a Krull monoid to be a bounded factorization monoid, and we provide arithmetical finiteness results in case of normalizing Krull monoids with finite Davenport constant.

The large Davenport constant II: General upper bounds

Let G be a finite group written multiplicatively. By a sequence over G, we mean a finite sequence of terms from G which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of G. The small Davenport constantd(G) is the maximal integer ℓ such that there is a sequence over G of length ℓ which has no nontrivial, product-one subsequence. The large Davenport constantD(G) is the maximal length of a minimal product-one sequence—this is a product-one sequence which cannot be partitioned into two nontrivial, product-one subsequences. The goal of this paper is to present several upper bounds for D(G), including the following: D(G)≤{d(G)+2|G′|−1,where G′=[G,G]≤G is the commutator subgroup;34|G|,if G is neither cyclic nor dihedral of order 2n with n odd;2p|G|,if G is noncyclic, where p is the smallest prime divisor of |G|;p2+2p−2p3|G|,if G is a non-abelian p-group.As a main step in the proof of these bounds, we will also show that D(G)=2q when G is a non-abelian group of order |G|=pq with p and q distinct primes such that p∣q−1.

The large Davenport constant I: Groups with a cyclic, index [formula omitted] subgroup

Let G be a finite group written multiplicatively. By a sequence over G, we mean a finite sequence of terms from G which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of G. The small Davenport constantd(G) is the maximal integer ℓ such that there is a sequence over G of length ℓ which has no nontrivial, product-one subsequence. The large Davenport constantD(G) is the maximal length of a minimal product-one sequence—this is a product-one sequence which cannot be factored into two nontrivial, product-one subsequences. It is easily observed that d(G)+1≤D(G), and if G is abelian, then equality holds. However, for non-abelian groups, these constants can differ significantly. Suppose G has a cyclic, index 2 subgroup. Then an old result of Olson and White (dating back to 1977) implies that d(G)=12|G| if G is non-cyclic, and d(G)=|G|−1 if G is cyclic. In this paper, we determine the large Davenport constant of such groups, showing that D(G)=d(G)+|G′|, where G′=[G,G]≤G is the commutator subgroup of G.

The set of distances in Krull monoids

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf c (H)$ of $H$ is the smallest integer $N$ with the following property: for each $a \in H$ and each two factorizations $z, z'$ of $a$, there exist factorizations $z = z_0, ..., z_k = z'$ of $a$ such that, for each $i \in [1, k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. Under a very mild condition on the Davenport constant of $G$, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between $\mathsf c (H)$ and the set of distances of $H$ and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on $\mathsf c(H)$ and characterize when $\mathsf c(H)\leq 4$.

New upper bounds for the Davenport and for the Erdős–Ginzburg–Ziv constants

Let G be a finite abelian group (written additively) of rank r with invariants n1, n2, . . . , nr, where nr is the exponent of G. In this paper, we prove an upper bound for the Davenport constant D(G) of G as follows; D(G) ≤ nr + nr-1 + (c(3) − 1)nr-2 + (c(4) − 1) nr-3 + · · · + (c(r) − 1)n1 + 1, where c(i) is the Alon–Dubiner constant, which depends only on the rank of the group \({{\mathbb Z}_{n_r}^i}\). Also, we shall give an application of Davenport’s constant to smooth numbers related to the Quadratic sieve.

Generalizations of some zero sum theorems

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory

Let D be the ring of integers in a finite extension of the rationals. The classic examination of the factorization properties of algebraic integers usually begins with the study of norms. In this paper, we show using the ideal class group, C(D), of D that a deeper examination of such properties is possible. Using the class group, we construct an object known as a block monoid, which allows us to offer proofs of three major results from the theory of nonunique factorizations: Geroldinger's theorem, Carlitz's theorem, and Valenza's theorem. The combinatorial properties of block monoids offer a glimpse into two widely studied constants from additive number theory, the Davenport constant and the cross number. Moreover, block monoids allow us to extend these results to the more general classes of Dedekind domains and Krull domains.

Zero-sum problems with congruence conditions

For a finite abelian group G and a positive integer d, let s dℕ(G) denote the smallest integer ℓ∈ℕ0 such that every sequence S over G of length |S|≧ℓ has a nonempty zero-sum subsequence T of length |T|≡0 mod d. We determine s dℕ(G) for all d≧1 when G has rank at most two and, under mild conditions on d, also obtain precise values in the case of p-groups. In the same spirit, we obtain new upper bounds for the Erdős–Ginzburg–Ziv constant provided that, for the p-subgroups G p of G, the Davenport constant D(G p ) is bounded above by 2exp (G p )−1. This generalizes former results for groups of rank two.

The Inverse Problem Associated to the Davenport Constant for $C_2\oplus C_2 \oplus C_{2n}$, and Applications to the Arithmetical Characterization of Class Groups

The inverse problem associated to the Davenport constant for some finite abelian group is the problem of determining the structure of all minimal zero-sum sequences of maximal length over this group, and more generally of long minimal zero-sum sequences. Results on the maximal multiplicity of an element in a long minimal zero-sum sequence for groups with large exponent are obtained. For groups of the form $C_2^{r-1}\oplus C_{2n}$ the results are optimal up to an absolute constant. And, the inverse problem, for sequences of maximal length, is solved completely for groups of the form $C_2^2 \oplus C_{2n}$. Some applications of this latter result are presented. In particular, a characterization, via the system of sets of lengths, of the class group of rings of algebraic integers is obtained for certain types of groups, including $C_2^2 \oplus C_{2n}$ and $C_3 \oplus C_{3n}$; and the Davenport constants of groups of the form $C_4^2 \oplus C_{4n}$ and $C_6^2 \oplus C_{6n}$ are determined.

The catenary degree of Krull monoids I

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree c(H) of H is the smallest integer N with the following property: for each a 2 H and each two factorizations z, z 0 of a, there exist factorizations z = z0, . . . , zk = z 0 of a such that, for each i 2 (1, k), zi arises from zi−1 by replacing at most N atoms from zi−1 by at most N new atoms. Under a very mild condition on the Davenport constant of G, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between c(H) and the set of distances of H and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on c(H) and characterize when c(H) � 4.

On the Olson and the Strong Davenport constants

Soit S un sous-ensemble d’un groupe abélien fini, noté additivement. Si 0 n’est pas une sous-somme (non vide) de S, on dit que S est un ensemble sans sous-somme nulle. Nous examinons la cardinalité maximale d’un ensemble sans sous-somme nulle, c’est-à-dire la (petite) constante d’Olson. Nous déterminons la cardinalité maximale d’un tel ensemble pour plusieurs types de groupes ; en particulier, les p-groupes dont le rang est suffisament grand relativement à l’exposant et plus particulièrement tous les groupes dont l’exposant est au plus 5. Nous obtenons ces résultats comme des cas particuliers de résultats plus généraux, donnant des bornes inférieures pour la cardinalité d’un ensemble sans sous-somme nulle pour des groupes variés. Nous examinons la qualité de ces bornes en considérant des cas explicites, avec l’aide d’un ordinateur. De plus, nous examinons une notion très proche de la constante d’Olson : la cardinalité maximale d’un ensemble minimal de somme nulle, c’est-à-dire la constante de Davenport forte. En particulier, nous déterminons la valeur de cette constante pour les p-groupes élémentaires dont le rang est au plus 2, en utilisant des résultats récents sur la constante d’Olson.

An application of coding theory to estimating Davenport constants

We investigate a certain well-established generalization of the Davenport constant. For j a positive integer (the case j = 1, is the classical one) and a finite Abelian group (G, +, 0), the invariant D j (G) is defined as the smallest ℓ such that each sequence over G of length at least ℓ has j disjoint non-empty zero-sum subsequences. We investigate these quantities for elementary 2-groups of large rank (relative to j). Using tools from coding theory, we give fairly precise estimates for these quantities. We use our results to give improved bounds for the classical Davenport constant of certain groups.

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 D k ( G ) denote the smallest ℓ such that each sequence over G of length at least ℓ 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 ( D k ( G ) ) k ∈ N is eventually an arithmetic progression with difference exp ( G ) , and several questions arising from this fact are investigated. For elementary 2 -groups, D k ( G ) 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).