On the zero-sum constant, the Davenport constant and their analogues

TL;DR

This paper explores the properties of zero-sum constants in finite Abelian groups, generalizes known relations, investigates asymptotic behaviors, and extends some results to non-Abelian groups, with applications to smooth numbers.

Contribution

It proves a general formula for ${ m E}_m(G)$ in Abelian groups, extends the relation to non-Abelian groups, and examines asymptotic behaviors of related sequences.

Findings

${ m E}_m(G)=D(G)-1+m|G|$ for Abelian groups

Generalization of Kemnitz's conjecture

Stronger version of a result by Delorme, Ordaz, Quiroz

Abstract

Let $D(G)$ be the Davenport constant of a finite Abelian group $G$. For a positive integer $m$ (the case $m = 1$, is the classical one) let ${\mathsf E}_m(G)$ (or $\eta_m(G)$, respectively) be the least positive integer $t$ such that every sequence of length $t$ in $G$ contains $m$ disjoint zero-sum sequences, each of length $|G|$ (or of length $\le exp(G)$ respectively). In this paper, we prove that if $G$ is an~Abelian group, then ${\mathsf E}_m(G)=D(G)-1+m|G|$, which generalizes Gao's relation. We investigate also the non-Abelian case. Moreover, we examine the asymptotic behavior of the sequences $({\mathsf E}_m(G))_{m\ge 1}$ and $(\eta_m(G))_{m\ge 1}.$ We prove a~generalization of Kemnitz's conjecture. The paper also contains a result of independent interest, which is a stronger version of a result by Ch. Delorme, O. Ordaz, D. Quiroz. At the and we apply the Davenport constant to smooth numbers and make a natural conjecture in the non-Abelian case.

Equations

$$\prod _{g\in G}{g}^{{\nu }_g(S)}=g_1\cdot \dots \cdot g_l,$$

$$\sigma (S)=\sum _{g\in G}{\nu }_g(S)g\in G.$$

$$D^{\ast }(G)\le D(G)\le n_r\left(1+\log \frac{|G|}{n_r}\right).$$

$$\frac{D(G)}{D^{\ast }(G)}\le (Kr\log r{)}^r,$$

$$s_{\mathbb{N}}(G)=s_{[1,D(G)]}(G)=D(G)=D_1(G),$$

$$s_{I,1}(G)=s_I(G),s_{[1,D(G)],n}(G)=D_n(G),s_{\{|G|\},n}=E_n(G).$$

$$s_{[1,\exp (G)],n}(G)\ge s_{[1,k^{\prime }],n}(G)\ge s_{[1,k],n}(G)\ge D_n(G).$$

$$s_{\le k^{\prime }}(G)\ge s_{\le k}(G).$$

$$E(G)=D(G)+|G|-1,$$

$$E_m(G)=E(G)+(m-1)|G|=D(G)+m|G|-1.$$

$$T=a_1\cdot a_2\cdot \dots \cdot a_{{}_{D(G)-1}}\cdot \underset{m|G|-1\mathrm{times}}{\underbrace{0\cdot \dots \cdot 0}}.$$

$$E_m(G)=m{p}^{{\sum }_{i=1}^k{e}_i}+\sum _{i=1}^k(p^{e_i}-1).$$

$$\eta (G)+m\exp (G)-1\le s_{(m)}(G)\le s(G)+(m-1)\exp (G).$$

$$T=a_1\cdot a_2\cdot \dots \cdot a_{{}_{\eta (G)-1}}\cdot \underset{m\exp (G)-1\mathrm{times}}{\underbrace{0\cdot \dots \cdot 0}},$$

$$E_m(G)=m{n}_2{n}_1+n_2+n_1-2,$$

$$D_m(G)=m{n}_2+n_1-1,$$

$$s_{(m)}(G)=(m+1)n_2+2n_1-3,$$

$$s_{(m)}(G)-D_m(G)=D(G)-1.$$

$$s_{\le (n-1)p}(C_p^n)\le (n+1)p-n.$$

$$s_{\le 2p}(C_p^3)\le 4p-3.$$

$$D_n(G)\le s_{[1,k],n}(G)\le s_{\le k}(G)+k(n-1),$$

$$D_n(G)\le s_{[1,\exp (G)],n}(G)\le \eta (G)+\exp (G)(n-1).$$

$$D(H)+D(G/H)-1\le D(G)\le D_{D(H)}(G/H)\le D(H)D(G/H).$$

$$n_3+n_2+p-2\le D(C_p\oplus C_{n_2}\oplus C_{n_3})\le D_{\frac{n_2}{p}+\frac{n_3}{p}-1}(C_p^3)\le 2n_3+2n_2-3.$$

$$D(G)\le D_{D(H)}(G/H)=D_{\frac{n_2}{p}+\frac{n_3}{p}-1}(C_p^3).$$

$$\begin{array}{rl}D(G)& \le s_{\le 2p}(C_p^3)+2p(\frac{n_2}{p}+\frac{n_3}{p}-2)\le \\ & \le 2p(\frac{n_2}{p}+\frac{n_3}{p}-2)+(4p-3)=2n_3+2n_2-3.\end{array}$$

$$s_{\le 2^{\Omega (h)}l}(G)\le 2^{\Omega (h)}(2l+k+h)-3.$$

$$s_{\le 2^{\Omega (h)}l}(G)=s_{\le l}(K\oplus L)\le 2l+k-2=2^{\Omega (1)}(2l+k+1)-3.$$

$$s_{\le 2^{\Omega (h_1)}{l}_1}(Q)\le 2^{\Omega (h_1)}(2l_1+k_1+h_1)-3.$$

$$s=2p\big{(}2^{\Omega(h_{1})}(2l_{1}+k_{1}+h_{1})-2\big{)}+4p-3.$$

Full text

††2010 Mathematics Subject Classification: Primary 11P70; Secondary 11B50.††Key words and phrases: zero-sum sequence, Davenport constant, finite abelian group

1 Introduction

We will define and investigate some generalizations of the Davenport constant (see [2, 11, 13, 16, 17, 27, 28, 30]). Davenport’s constant is connected with algebraic number theory as follows. For an algebraic number field $K$, let $\mathcal{O}_{K}$ be its ring of integers and $G$ the ideal class group of $\mathcal{O}_{K}$. Let $x\in\mathcal{O}_{K}$ be an irreducible element. If $\mathcal{O}_{K}$ is a Dedekind domain, then $x\mathcal{O}_{K}=\prod\limits_{i=1}^{r}\mathcal{P}_{i}$, where $\mathcal{P}_{i}$ are prime ideals in $\mathcal{O}_{K}$ (not necessarily distinct). The Davenport constant $D(G)$ is the maximal number of prime ideals $\mathcal{P}_{i}$ (counted with multiplicites) in the prime ideal decomposition of the integral ideal $x\mathcal{O}_{K}$ (see [23, 27]).

The precise value of the Davenport constant is known, among others, for $p$-groups and for groups of rank at most two. The determination of $D(G)$ for general finite Abelian groups is an open question (see [20]).

2 General notation

Let $\mathbb{N}$ denote the set of positive integers (natural numbers). We set

$[a,b]=\{x:a\leq x\leq b,\,\,x\in\mathbb{Z}\},$ where $a,b\in\mathbb{Z}.$ Our notation and terminology is consistent with [19]. Let $G$ be a non-trivial additive finite Abelian group. $G$ can be uniquely decomposed as a direct sum of cyclic groups $C_{n_{1}}\oplus C_{n_{2}}\oplus\ldots\oplus C_{n_{r}}$ with natural numbers $1<n_{1}|\ldots|n_{r}.$ The number $r$ of summands in the above decomposition of $G$ is expressed as $r=r(G)$ and called the rank of $G.$ The integer $n_{r}$ is called the exponent of $G$ and denoted by $\exp(G).$ In addition, we define $D^{*}(G)$ as $D^{*}(G)=1+\sum\limits_{i=1}^{r}(n_{i}-1)$.

We write any finite sequence $S$ of $l$ elements of $G$ in the form

[TABLE]

(this is a formal Abelian product), where $l$ is the length of $S$ denoted by $|S|,\nu_{g}(S)$ is the multiplicity of $g$ in $S.$ $S$ corresponds to the sequence (in the traditional sense) $(g_{1},g_{2},\ldots,g_{l})$ where we forget the ordering of the terms. By $\sigma(S)$ we denote the sum of $S:$

[TABLE]

The Davenport constant $D(G)$ is defined as the smallest $t\in\mathbb{N}$ such that each sequence over $G$ of length at least $t$ has a non-empty zero-sum subsequence. Equivalently, $D(G)$ is the maximal length of a zero-sum sequence of elements of $G$ and with no proper zero-sum subsequence. One of the best bounds for $D(G)$ known so far is:

[TABLE]

Alford, Granville and Pomerance [1] in 1994 used the bound (1), to prove the existence of infinitely many Carmichael numbers. Dimitrov [8] used the Alon Dubiner constant (see [2]) to prove the inequality:

[TABLE]

for an absolute constant $K$. It is known that for groups of rank at most two and for $p$-groups, where $p$ is a prime, the left hand side inequality (1) is in fact an equality (see Olson [27]). This result suggests that $D^{*}(G)=D(G).$ However, there are infinitely many groups $G$ with rank $r>3$ such that $D(G)>D^{*}(G)$. There are more recent results on groups where the Davenport constant does not match the usual lower bound (see [18]). The following Remark 2.1 lists some basic facts for the Davenport constant (see [7, 18]).

Remark 2.1**.**

Let $G$ be a finite additive Abelian group.

Then $D(G)=D^{*}(G)$ in each of the following cases:

1. (a)

$G$ is a $p-$group, 2. (b)

$G$ has rank $r\leq 2,$ 3. (c)

$G=C_{p}\oplus C_{p}\oplus C_{p^{n}m},$ with $p$ a prime number, $n\geq 2$ and $m$ an integer coprime with $p^{n}$ (more generally, if $G=G_{1}\oplus C_{p^{k}n},$ where $G_{1}$ is a $p-group$ and $p^{k}\geq D^{*}(G_{1})),$ 4. (d)

$G=C_{2}\oplus C_{2}\oplus C_{2}\oplus C_{2n},$ with odd $n,$ 5. (e)

$G=C_{2p^{k_{1}}}\oplus C_{2p^{k_{2}}}\oplus C_{2p^{k_{3}}},$ with $p$ prime, 6. (f)

$G=C_{2}\oplus C_{2n}\oplus C_{2nm},$ 7. (g)

$G=C_{3}\oplus C_{3n}\oplus C_{3nm}.$ 2. 2.

Then $D(G)>D^{*}(G)$ in each of the following cases:

1. (a)

$G=C_{n}\oplus C_{n}\oplus C_{n}\oplus C_{2n}$ for every odd $n\geq 3,$ 2. (b)

$G=C_{3}\oplus C_{9}\oplus C_{9}\oplus C_{18},$ 3. (c)

$G=C_{3}\oplus C_{15}\oplus C_{15}\oplus C_{30}.$ 4. (d)

Let $n\geq 2,\,k\geq 2,\,(n,k)=1,0\leq\rho\leq n-1,$ and

$G=C_{n}^{(k-1)n+\rho}\oplus C_{kn}.$ If $\rho\geq 1$ and $\rho\not\equiv n\pmod{k}$, then $D(G)\geq D^{*}(G)+\rho.$ If $\rho\leq n-2$ and $x(n-1-\rho)\not\equiv n\pmod{k}$

for any $x\in[1,n-1],$ then $D(G)\geq D^{*}(G)+\rho+1.$

3 Definitions

In this section, we will provide some definitions of classical invariants. We begin with some notation and remarks that will be used throughout the paper.

Definition 3.1**.**

Let $G$ be a finite Abelian group, $m$ and $k$ positive integers such that $k\geq\exp(G),$ and $\emptyset\neq I\subset\mathbb{N}.$

1. By $s_{I}(G)$ we denote the smallest $t\in\mathbb{N}\cup\{\infty\}$ such that every sequence $S$ over $G$ of length $t$ contains a non-empty subsequence $S^{\prime}$ such that $\sigma(S^{\prime})=0$, $|S^{\prime}|\in I$ (see [3, 6, 7]).

We use notation $s_{\leq k}(G)$ or $D^{k}(G)$ to denote $s_{I}(G)$ if $I=[1,k]$ (see [3, 7]).

2. By $s_{I,m}(G)$ we denote the smallest $t\in\mathbb{N}\cup\{\infty\}$ such that every sequence $S$ over $G$ of length $t$ contains at least $m$ disjoint non-empty subsequences $S_{1},S_{2},\ldots,S_{m}$ such that $\sigma(S_{i})=0$, $|S_{i}|\in I$.

3. Let ${\mathsf{E}}(G):=s_{\{|G|\}}(G),$ i.e. the smallest $t\in\mathbb{N}\cup\{\infty\}$ such that every sequence $S$ over $G$ with length $t$ contains non-empty subsequence $S^{\prime}$ such that $\sigma(S^{\prime})=0,\,|S^{\prime}|=|G|.$

Note that ${\mathsf{E}}(G)$ is the classical zero-sum constant.

4. Let ${\mathsf{E}}_{m}(G):=s_{\{|G|\},m}(G),$ i.e. the smallest $t\in\mathbb{N}\cup\{\infty\}$ such that every sequence $S$ over $G$ with length $t$ contains at least $m$ disjoint non-empty subsequences $S_{1},S_{2},\ldots,S_{m}$ such that $\sigma(S_{i})=0,\,|S_{i}|=|G|,$ for $i\in[1,m].$

5. Also, we define $\eta(G):=s_{\leq\exp(G)},\,s(G):=s_{\{\exp(G)\}}(G),\,D_{m}(G):=s_{\mathbb{N},m}(G)$ (see [23]), $s_{(m)}(G):=s_{\{\exp(G)\},m}(G).$

Remark 3.2**.**

For $n,n^{\prime}\in\mathbb{N},\,\emptyset\neq I\subseteq\mathbb{N},$ by definition

[TABLE]

[TABLE]

Note that $s(G)=s_{\{exp(G)\}}(G)$ is the classical Erdös - Ginzburg - Ziv constant (see [12]). We call $s_{(m)}(G)=s_{\{\exp(G)\},m}(G)$ the $m$-wise Erdös–Ginzburg–Ziv constant of $G.$ Thus, in this notation $s_{(1)}(G)=s(G).$

One can derive, for example, the following inequalities.

Remark 3.3**.**

If $n^{\prime}\geq n$, then $s_{I,n^{\prime}}(G)\geq s_{I,n}(G).$

If $D(G)\geq k\geq k^{\prime}\geq\exp(G)$, then

[TABLE]

If $D(G)\geq k\geq\exp(G)$, then $\eta(G)\geq s_{\leq k}(G)\geq D(G).$

If $k\geq k^{\prime}$, then

[TABLE]

We note that a sequence $S$ over $G$ of length $|S|\geq mD(G)$ can be partitioned into $m$ disjoint subsequences $S_{i}$ of length $|S_{i}|\geq D(G).$ Thus, each $S_{i}$ contains a non-empty zero-sum subsequence. Hence, $D_{m}(G)\leq mD(G).$ See also [23] Proposition 1 (ii).

4 The $m$–wise zero-sum constant and the $m$–wise Erdös–Ginzburg–Ziv constant of $G$.

In 1996, Gao and Caro proved independently that:

[TABLE]

for any finite Abelian group (see [15, 5]]; for a proof in modern language we refer to [17, Proposition 5.7.9]; see also [14, 7, 24, 31]). Relation (3) unifies research on constants $D(G)$ and $E(G).$ We start this section with the result that can be used to unify research on constants $D(G)$ and $E_{m}(G).$

Theorem 4.1**.**

If $G$ is a finite Abelian group of order $|G|$, then

[TABLE]

Proof.

By (3) we obtain $E(G)+(m-1)|G|=D(G)+m|G|-1$.

Let $S=g_{1}g_{2}\cdot\ldots\cdot g_{{}_{D(G)-1}}$ be a sequence of $D(G)-1$ non-zero elements in $G.$ Using the definition of $D(G)$, we may assume that $S$ does not contain any non-empty subsequence $S^{\prime}$ such that $\sigma(S^{\prime})=0$. We put

[TABLE]

We observe that the sequence $T$ dose not contain $m$ disjoint non-empty subsequences $T_{1},T_{2},\ldots,T_{m}$ such that $\sigma(T_{i})=0$ and $|T_{i}|=|G|$ for $i\in[1,m].$ This implies that $E_{m}(G)>D(G)+m|G|-2.$ Hence, $E_{m}(G)\geq E(G)+(m-1)|G|.$ On the other hand, if $S$ is any sequence such that $|S|\geq E(G)+(m-1)|G|,$ then one can sequentially extract at least $m$ disjoint subsequences $S_{1},\ldots,S_{m},$ such that $\sigma(S_{i})=0$ in $G$ and $|S_{i}|=|G|.$ Thus, $E_{m}(G)\leq E(G)+(m-1)|G|$. ∎

We anticipate that Theorem 4.1 can be used to obtain a generalization of the classical Theorem of Hall (see [26, Section 3]).

Corollary 4.2**.**

For every finite Abelian group the sequence $({\mathsf{E}}_{m}(G))_{m\geq 1}$ is an arithmetic progression with difference $|G|.$

Corollary 4.3**.**

If $p$ is a prime and $G=C_{{}_{p^{e_{1}}}}\oplus\ldots\oplus C_{{}_{p^{e_{k}}}}$ is a $p-$group, then for a natural $m$ we have:

[TABLE]

Proof.

It follows from Remark 2.1 and Theorem 4.1. ∎

We recall that by $s_{(m)}(G)$ we denote the smallest $t\in\mathbb{N}\cup\{\infty\}$ such that every sequence $S$ over $G$ of length $t$ contains at least $m$ disjoint non-empty subsequences $S_{1},S_{2},\ldots,S_{m}$ such that $\sigma(S_{i})=0,\,|S_{i}|=\exp(G)$.

Theorem 4.4**.**

If $G$ is a finite Abelian group, then

[TABLE]

Proof.

The proof runs along the same lines as the proof of Theorem 4.1.

Let $S=g_{1}g_{2}\cdot\ldots\cdot g_{{}_{\eta(G)-1}}$ be a sequence of $\eta(G)-1$ non-zero elements in $G.$ Using the definition of $\eta(G)$, we may assume that $S$ does not contain any non-empty subsequence $S^{\prime}$ such that $\sigma(S_{i})=0,\,|S_{i}|\leq\exp(G)$. We put

[TABLE]

We observe that the sequence $T$ not contain $m$ disjoint non-empty subsequences $T_{1},T_{2},\ldots,T_{m}$ such that $\sigma(T_{i})=0$ and $|T_{i}|=\exp(G)$ for

$i\in[1,m].$ This implies that $s_{(m)}(G)>\eta(G)+m\exp(G)-2.$ Hence,

$s_{(m)}(G)\geq\eta(G)+m\exp(G)-1.$ On the other hand, if $S$ is any sequence over $G$ such that $|S|\geq s(G)+(m-1)|G|,$ then one can sequentially extract at least $m$ disjoint subsequences $S_{1},\ldots,S_{m},$ such that $\sigma(S_{i})=0$ in $G$ and $|S_{i}|=\exp(G).$ Thus, $s_{(m)}(G)\leq s(G)+(m-1)\exp(G).$ ∎

It was conjectured by Gao that for every finite Abelian group $G,$ one has $\eta(G)+\exp(G)-1=s(G)$ (see [16, Conjecture 6.5]). If this conjecture is true, then by Theorem 4.4 for every finite Abelian group $G$ the equality $s_{(m)}(G)=s(G)+(m-1)\exp(G)$ holds, i.e. the sequence $(s_{(m)}(G))_{m\geq 1}$ is an arithmetic progression with difference $\exp(G).$ We will note that the equation $\eta(G)+\exp(G)-1=s(G)$ is true for all finite Abelian groups of rank at most two (see [21, Theorem 2.3]).

Corollary 4.5**.**

If $G=C_{n_{1}}\oplus C_{n_{2}},$ where $n_{1}|n_{2},$ then for a natural $m$ we have:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

The equation (5) is a consequence of Remark 2.1 and Theorem 4.1, the equation (6) follows from Proposition 5 in [23]. By applying [21, Theorem 2.3] and Theorem 4.4, we can obtain (7). The equation (8) is a consequence of equations (6) and (7). Note that, if $G$ is an Abelian group, then $D(G)-1$ is the maximal length of a zero-sum free sequence over $G.$ ∎

5 A generalization of the Kemnitz conjecture

Kemnitz’s conjecture states that every set $S$ of $4n-3$ lattice points in the plane has a subset $S^{\prime}$ with $n$ points whose centroid is also a lattice point. The conjecture was proved by Christian Reiher (see [29]). In order to prove the generalization of this theorem, we will use the equation (7).

Theorem 5.1**.**

*Let $n$ and $m$ be natural numbers. Let $S$ be a set of

$(m+3)n-3$ lattice points in $2$-dimensional Euclidean space. Then there are at least $m$ pairwise disjoint sets $S_{1},S_{2},\ldots,S_{m}\subseteq S$ with $n$ points each, such that the centroid of each set $S_{i}$ is also a lattice point.*

Proof.

As Harborth has already noted (see [11, 25]), $s(C_{n}^{r})$ is the smallest integer $l$ such that every set of $l$ lattice points in $r$-dimensional Euclidean space contains $n$ elements which have a centroid in a lattice point. By analogy, $s_{(m)}(C_{n}^{r})$ is the smallest integer $l$ such that every set $S$ of $l$ lattice points in $r$-dimensional Euclidean space have $m$ pairwise disjoint subsets $S_{1},S_{2},\ldots,S_{m}$ each of cardinality $n,$ which centroids are also a lattice points.

Finally, in the case of $2$-dimensional Euclidean space, it is sufficient to use the equation (7). ∎

Remark 5.2**.**

As we can see in point 3) of Definition 3.1, Theorem 5.1 is also true when we replace sets with multisets.

6 Some results on $s_{I,m}(G)$ constant

In this section, we will investigate zero-sum constants for finite Abelian groups. We start with $s_{\leq k}(G),D_{m}(G),\eta(G)$ constants. Our main result of this section is Theorem 6.11. Olson calculated $s_{\leq p}(C_{p}^{2})$ for a prime number $p$ (see [28]). No precise result is known for $s_{\leq p}(C_{p}^{n}),$ where $n\geq 3.$ We need two technical lemmas:

Lemma 6.1**.**

Let $p$ be a prime number and $n\geq 2.$ Then:

[TABLE]

Proof.

Let $g_{i}\in C_{p}^{n},\,i\in[1,(n+1)p-n].$ Embed $C_{p}^{n}$ into an Abelian group $F$ group isomorphic to $C_{p}^{n+1}.$ Let $x\in F,\,x\notin C_{p}^{n}.$ Since $D(C_{p}^{n+1})=(n+1)p-n$ (see [27] or Remark 2.1,(a)) there exists a zero-sum subsequence $\prod\limits_{i\in I}(x+g_{i})$ of the sequence $\prod\limits_{i=1}^{(n+1)p-n}(x+g_{i}).$ But this possible only if $p$ divides $|I|.$ Rearranging subscripts, we may assume that $g_{1}+g_{2}+\ldots+g_{ep}=0,$ where $e\in[1,n].$ We are done if $e\in[1,n-1].$ If $e=n$ we obtain a zero-sum sequence $S=g_{1}\cdot g_{2}\cdot\ldots\cdot g_{np}.$ Zero-sum sequence $S$ contains a proper zero-sum subsequence $S^{\prime},$ since $D(C_{p}^{n})=np-(n-1),$ and thus zero-sum subsequence of length not exceeding $\lfloor\frac{np+1}{2}\rfloor\leq(n-1)p.$ ∎

Corollary 6.2**.**

Let $p$ be a prime. Then:

[TABLE]

Lemma 6.3**.**

*Let $G$ be a finite Abelian group, $k\in\mathbb{N},\,k\geq\exp(G)$.

If $s_{[1,k],1}(G)\leq s_{[1,k],m}(G)+k$, then $s_{[1,k],m+1}(G)\leq s_{[1,k],m}(G)+k.$*

Proof.

Let $S$ be a sequence over $G$ of length $s_{[1,k],m}(G)+k$. The sequence $S$ contains a non-empty subsequence $S_{0}|S$ such that $\sigma(S_{0})=0.$ Then $|S_{0}|\in[1,k],$ since $|S|\geq s_{[1,k],1}(G)=s_{\leq k}(G).$ By definition of $s_{[1,k],m}(G)$ the remaining elements in $S$ contain $m$ disjoint non-empty subsequences $S_{i}|S$ such that $\sigma(S_{i})=0$, $|S_{i}|\in[1,k],$ where $i\in[1,m].$ Thus, we get $m+1$ non-empty disjoint subsequences $S_{i}|S$ such that $\sigma(S_{i})=0$, $|S_{i}|\in[1,k],$ where $i\in[0,m].$ ∎

Corollary 6.4**.**

*Let $G$ be a finite Abelian group, $k\geq\exp(G).$

If $s_{[1,k],1}(G)\leq s_{[1,k],m}(G)+k,$ then $s_{[1,k],m+n}(G)\leq s_{[1,k],m}(G)+nk.$*

Proof.

We use Lemma 6.3 and Remark 3.2. ∎

Corollary 6.5**.**

Let $G$ be a finite Abelian group, $k\geq\exp(G)$. Then:

[TABLE]

[TABLE]

Proof.

We use Remark 3.3, Corrolary 6.4 with $m=1$ and get (11).

We put $k=\exp(G)$ in (11) and get (12). ∎

Remark 6.6**.**

It is known that:

$\eta(C_{n}^{3})=8n-7,$ if $n=3^{\alpha}5^{\beta},$ with $\alpha,\beta\geq 0;$

$\eta(C_{n}^{3})=7n-6$, if $n=2^{\alpha}3$, with $\alpha\geq 1;$

$\eta(C_{2}^{3})=8;\eta(C_{3}^{3})=17;\eta(C_{3}^{4})=39;\eta(C_{3}^{5})=89;\eta(C_{3}^{6})=223,$ (see [20]).

Corollary 6.7**.**

*We have that:

$D_{m}(C_{n}^{3})\leq nm+7n-7,$ if $n=3^{\alpha}5^{\beta},$ with $\alpha,\beta\geq 0;$

$D_{m}(C_{n}^{3})\leq nm+6n-6$, if $n=2^{\alpha}3$, with $\alpha\geq 1;$

$D_{m}(C_{3}^{4})\leq 3m+36;D_{m}(C_{3}^{5})\leq 3m+86;D_{m}(C_{3}^{6})\leq 3m+220.$*

Proof.

By Corollary 6.5 and Remark 6.6. ∎

In the next Lemma we collect several useful properties on the Davenport constant.

Lemma 6.8**.**

Let $G$ be a non-trivial finite Abelian group and $H$ subgroup of $G.$ Then:

[TABLE]

Proof.

The inequality $D(H)+D(G/H)-1\leq D(G)$ is proved in Proposition 3(i) in [23]. Now we prove the inequality $D(G)\leq D_{D(H)}(G/H)$ on the same lines as in [7] (we include the proof for the sake of completeness). If $|S|\geq D_{D(H)}(G/H)$ is any sequence over $G$, then one can, by definition, extract at least $D(H)$ disjoint non empty subsequences $S_{1},\ldots,S_{D(H)}|S$ such that $\sigma(S_{i})\in H.$ Since $T=\prod\limits_{i=1}^{D(H)}\sigma(S_{i})$ is a sequence over $H$ of length $D(H),$ thus there exists a non-empty subset $I\subseteq[1,D(H)]$ such that $T^{\prime}=\prod\limits_{i\in I}\sigma(S_{i})$ is a zero-sum subsequent of $T.$

We obtain that $S^{\prime}=\prod\limits_{i\in I}S_{i}$ is a non-empty zero-sum subsequence of $S$.

The inequality $D_{D(H)}(G/H)\leq D(H)D(G/H)$ follows from Remark 3.3. ∎

Theorem 6.9**.**

For an Abelian group $C_{p}\oplus C_{n_{2}}\oplus C_{n_{3}}$ such that $p|n_{2}|n_{3}\in\mathbb{N},$ where $p$ is a prime number, we have:

[TABLE]

Proof.

If $G=C_{p}\oplus C_{n_{2}}\oplus C_{n_{3}}$ such that $p|n_{2}|n_{3}\in\mathbb{N},$ then $\exp(G)=n_{3}.$ Note that: $n_{3}+n_{2}+p-2=D^{*}(G)\leq D(G).$ Denoting by $H$ a subgroup of $G$ such that $H\cong C_{\tfrac{n_{2}}{p}}\oplus C_{\tfrac{n_{3}}{p}}.$ The quotient group $G/H\cong C_{p}\oplus C_{p}\oplus C_{p}.$

By Lemma 6.8 we get

[TABLE]

By (11) (with $m={\tfrac{n_{2}}{p}+\tfrac{n_{3}}{p}-1},\,k=2p$) and Corollary 6.2, we get

[TABLE]

∎

Our next goal is to generalize [7, Theorem 3.2] to the case $r(G)\geq 3.$

Theorem 6.10**.**

*Let $H,K$ and $L$ be Abelian groups of orders $|H|=h,$

$|K|=k$ and $|L|=l$. If $G=H\oplus K\oplus L$ with $h|k|l$.

Let $\Omega(h)$ denote the total number of prime factors of $h.$ Then:*

[TABLE]

Proof.

The proof will be inductive. If $h=1,$ then by [7, Theorem 3.2] we have:

[TABLE]

Assume that $h>1$ and let $p$ be a prime divisor of $h.$

Let $H_{1}$ be a subgroup of $H,$ $K_{1}$ be a subgroup of $K,$ $L_{1}$ be a subgroup of $L,$ with indices $[H:H_{1}]=[K:K_{1}]=[L:L_{1}]=p.$

Put $h=ph_{1},\,k=pk_{1},\,l=pl_{1}$ and $Q=H_{1}\oplus K_{1}\oplus L_{1}.$

Assume inductively that theorem is true for $Q$ i.e.

[TABLE]

Let $s=2^{\Omega(h)}(2l+k+h)-3$ and $S=g_{1}g_{2}\cdot\ldots\cdot g_{s}$ be a sequence of $G.$

We shall prove that there exists a subsequence of $S$ with length smaller or equal to $2^{\Omega(h)}l$ and zero sum. Let $b_{i}=g_{i}+Q\in G/Q.$ We consider the sequence $\prod\limits_{i=1}^{s}b_{i}$ of length $s.$

The quotient group $G/Q$ is isomorphic to $C_{p}^{3}$ and

[TABLE]

Therefore, by Lemma 6.2 there exist pairwise disjoint sets $I_{j}\subset[1,s],$

$|I_{j}|\leq~{}2p$ and

[TABLE]

such that each sequence $\prod\limits_{i\in I_{j}}b_{i}$ has a zero sum in $G/Q.$

In other words $\sigma(\prod\limits_{i\in I_{j}}g_{i})=\sum\limits_{i\in I_{j}}g_{i}\in Q.$ By induction assumption for $Q$ there exists $J\subseteq[1,j_{0}]$ with $|J|\leq 2^{\Omega(h_{1})}l_{1}$ such that $\sum\limits_{j\in J}\sigma(\prod\limits_{i\in I_{j}}g_{i})=0.$ Thus, we obtain zero sum subsequence of $S$ in $G$ of length not exceeding $\sum\limits_{j\in J}|I_{j}|\leq 2^{\Omega(h_{1})}l_{1}\cdot 2p=2^{\Omega(h)}l,$ which ends the inductive proof. ∎

Theorem 6.11**.**

*Let $H_{1},H_{2},\ldots,H_{n}$ be Abelian groups of orders $|H_{i}|=h_{i}$.

If $n\geq 2$ and $G=H_{1}\oplus H_{2}\oplus\ldots\oplus H_{n}$ with $h_{1}|h_{2}|\ldots|h_{n}$, then*

[TABLE]

Proof.

We proceed by induction on $n$ and $h_{n}.$

If $n=2$, then the inequality (22) holds by [[7] Theorem 3.2]. Namely:

[TABLE]

Suppose that the inequality (22) holds for fixed $n-1\geq 2$:

[TABLE]

If $n\geq 3$ and $h_{1}=1,$ then $G$ and $H_{2}\oplus\ldots\oplus H_{n}$ are isomorphic. By (2):

[TABLE]

Thus by the induction hypothesis (24):

[TABLE]

[TABLE]

Therefore (22) holds.

Suppose that the inequality (22) holds for fixed $n\geq 3$ and fixed $h,$ such that $h_{1}>h\geq 1$:

[TABLE]

Let $p$ be a prime divisor of $h_{1}.$ Let $H_{i}^{*}$ be a subgroup of index $p$ of a group $H_{i}$. Put $h_{i}=ph_{i}^{*}$ and $Q=H_{1}^{*}\oplus H_{2}^{*}\oplus\ldots\oplus H_{n}^{*}.$ By inductive assumption, the inequality (22) holds for $Q$:

[TABLE]

We put $s=(n-1)^{\Omega(h_{n})}(2(h_{n}-1)+(h_{n-1}-1)+\ldots+(h_{1}-1)+1)$ and let $S=g_{1}g_{2}\cdot\ldots\cdot g_{s}$ be a sequence of $G.$

We shall prove that there exists a subsequence of $S$ with length smaller or equal to $(n-1)^{\Omega(h_{n})}h_{n}$ and zero sum. Let $b_{i}=g_{i}+Q,\,1\leq i\leq s,\,$ be the sequence of $G/Q.$ The quotient group $G/Q$ is isomorphic to $C_{p}^{n}$ and

[TABLE]

Therefore, by Lemma 6.1 there exist pairwise disjoint $I_{j}\subseteq[1,s]$ with $|I_{j}|\leq(n-1)p$ and

[TABLE]

such that sequence $\prod\limits_{i\in I_{j}}b_{i}$ has a zero sum in $G/Q.$

In another words $\sigma(\prod\limits_{i\in I_{j}}g_{i})=\sum\limits_{i\in I_{j}}g_{i}\in Q.$ By induction (22) for $Q$ there exists $J\subseteq[1,j_{0}]$ with $|J|\leq(n-1)^{\Omega(h_{n}^{*})}h_{n}^{*}$ such that $\sum\limits_{j\in J}\sigma(\prod\limits_{i\in I_{j}}g_{i})=0.$

Thus, we obtain zero sum subsequence of $S$ in $G$ of length not exceeding $\sum\limits_{j\in J}|I_{j}|\leq(n-1)^{\Omega(h_{n}^{*})}h_{n}^{*}\cdot(n-1)p=(n-1)^{\Omega(h_{n})}h_{n}.$

∎

7 Davenport’s constant and smooth numbers

First, we recall the notion of a smooth number. Let $F=\{q_{1},q_{2},\ldots,q_{r}\}$ be a subset of positive integers. A positive integer $k$ is said to be smooth over a set $F$ if $k=q_{1}^{e_{1}}\cdot q_{2}^{e_{2}}\cdot\ldots\cdot q_{r}^{e_{r}}$ where $e_{i}$ are non-negative integers.

Remark 7.1**.**

Let $n\in\mathbb{N}$. Each smooth number over a set $\{{q_{1}}^{n},{q_{2}}^{n},\ldots,{q_{r}}^{n}\}$ is an $n$-th power of a suitable smooth number over the set $\{q_{1},q_{2},\ldots,q_{r}\}.$

Definition 7.2**.**

Let $\{p_{1},p_{2},\ldots,p_{r}\}$ be a set of distinct prime numbers. By $c(n_{1},n_{2},\ldots,n_{r}),$ we denote the smallest $t\in\mathbb{N}$ such that every sequence $M$ of smooth numbers over a set $\{p_{1},p_{2},\ldots,p_{r}\}$, of length $t$ has a non empty subsequence $N$ such that the product of all the terms of $N$ is a smooth number over a set $\{{p_{1}}^{n_{1}},{p_{2}}^{n_{2}},\ldots,{p_{r}}^{n_{r}}\}.$

Theorem 7.3**.**

Let $n_{1},n_{2},\ldots,n_{r}$ be integers such that $1<n_{1}|n_{2}|\ldots|n_{r}.$ Then:

[TABLE]

Proof.

It follows on the same lines as proof of [6, Theorem 1.6.]. First we will prove that

[TABLE]

We put $l=D(\mathbb{Z}_{n_{1}}\oplus\mathbb{Z}_{n_{2}}\oplus\ldots\oplus\mathbb{Z}_{n_{r}}).$ Let $M=(m_{1},m_{2},\ldots,m_{l})$ be a sequence of smooth numbers with respect to $F=\{p_{1},p_{2},\ldots,p_{r}\}$.

For all $i\in[1,n]$, we have $m_{i}=p_{1}^{e_{i,1}}p_{2}^{e_{i,2}}\cdot\ldots\cdot p_{r}^{e_{i,1}}$ where $e_{i,j}$ are non-negative integers. We associate each $m_{i}$ with $a_{i}\in\mathbb{Z}_{n_{1}}\oplus\mathbb{Z}_{n_{2}}\oplus\ldots\oplus\mathbb{Z}_{n_{r}}$ under the homomorphism:

[TABLE]

Hence:

[TABLE]

Thus, we get a sequence $S=a_{1}a_{2}\cdot\ldots\cdot a_{l}$ of elements of the group

$\mathbb{Z}_{n_{1}}\oplus\mathbb{Z}_{n_{2}}\oplus\ldots\oplus\mathbb{Z}_{n_{r}}$ of length $l=D(\mathbb{Z}_{n_{1}}\oplus\mathbb{Z}_{n_{2}}\oplus\ldots\oplus\mathbb{Z}_{n_{r}}).$ Therefore, there exists a non-empty zero sum subsequence $T$ of $S$ in $\mathbb{Z}_{n_{1}}\oplus\mathbb{Z}_{n_{2}}\oplus\ldots\oplus\mathbb{Z}_{n_{r}},$ and let $T=a_{j_{1}}a_{j_{2}}\cdot\ldots\cdot a_{j_{t}}.$ That is

[TABLE]

where $k\in[1,r].$ Consider the subsequence $N$ of $M$ corresponding to $T.$

We have $N=(m_{j_{1}},m_{j_{2}},\ldots,m_{j_{t}})$ and by equation (35), we get

[TABLE]

for some integers $l_{k}\geq 0.$ Hence $\prod\limits_{m\in N}m$ is a smooth number over a set $\{{p_{1}}^{n_{1}},{p_{2}}^{n_{2}},\ldots,{p_{r}}^{n_{r}}\}$. By definition of $c(n_{1},n_{2},\ldots,n_{r})$ we get inequality (32). To prove

[TABLE]

Let $l=c(n_{1},n_{2},\ldots,n_{r})$ and $S=a_{1}a_{2}\cdot\ldots\cdot a_{l}$ be a sequence of elements of $\mathbb{Z}_{n_{1}}\oplus\mathbb{Z}_{n_{2}}\oplus\ldots\oplus\mathbb{Z}_{n_{r}}$ of length $l,$ where a_{i}=\big{(}[e_{i,1}]_{{}_{n_{1}}},\,[e_{i,2}]_{{}_{n_{2}}},\,\ldots,\,[e_{i,r}]_{{}_{n_{r}}}\,\big{)}. We put $m_{i}=p_{1}^{e_{i,1}}\cdot p_{2}^{e_{i,2}}\cdot\ldots\cdot p_{r}^{e_{i,r}}.$ The sequence $M=(m_{1},m_{2},\ldots,m_{l})$ of integers is a sequence of smooth numbers over a set $F.$

By definition of $l=c(n_{1},n_{2},\ldots,n_{r}),$ there exists a non-empty subsequence $N=(m_{j_{1}},m_{j_{2}},\ldots,m_{j_{t}})$ of $M$ such that:

[TABLE]

for some integers $l_{k}\geq 0.$ The subsequence $T$ of $S$ corresponding to $N$ will sum up to the identity in $\mathbb{Z}_{n_{1}}\oplus\mathbb{Z}_{n_{2}}\oplus\ldots\oplus\mathbb{Z}_{n_{r}}.$ Therefore (37) holds and we obtain (31). ∎

8 Future work and the non-Abelian case

The constant $E(G)$ has received a lot of attention in the last $10$ years. For example, one direction went towards weighted zero-sum problems, see [22, Chapter 16]. The second direction went towards non-Abelian groups.

Let $G$ be any additive finite group. Let $S=(a_{1},\ldots,a_{n})$ be a sequence over $G$. We say that the sequence $S$ is a zero-sum sequence if there exists a permutation $\sigma:[1,n]\to[1,n]$ such that $0=a_{\sigma(1)}+\ldots+a_{\sigma(n)}.$ For a subset $I\subset\mathbb{N},$ let $s_{L}(G)$ denote the smallest $t\in\mathbb{N}\cup\{0,\infty\}$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a zero-sum subsequence $S^{\prime}$ of length $|S^{\prime}|\in L.$ The constants $D(G):=s_{\mathbb{N}}(G)$ and $E(G):=s_{\{|G|\}}(G)$ are classical invariants in zero-sum theory (independently of whether $G$ is Abelian or not). For a given finite group $G$, let ${\mathsf{d}}(G)d(G)$ be the maximal length of a zero-sum free sequence over $G.$ We call ${\mathsf{d}}(G)$ the small Davenport constant.

Remark 8.1**.**

In the Abelian case ${\mathsf{d}}(G)=D(G)-1$. Note that in non-Abelian groups ${\mathsf{d}}(G)$ can be strictly smaller than $D(G)-1$, see [19, Chapter 2].

Theorem 8.2**.**

If $G$ is a finite group of order $|G|$, then

[TABLE]

Proof.

Let $S=(g_{1},g_{2},\ldots,g_{{}_{d(G)}})$ be a sequence of $d(G)$ non-zero elements in $G.$ Using the definition of $d(G)$, we may assume that $S$ does not contain any non-empty subsequence $S^{\prime}$ such that $\sigma(S^{\prime})=0$. We put

[TABLE]

where $\nu_{0}(T)=m|G|-1.$

We observe that the sequence $T$ not contain $m$ disjoint non-empty subsequences $T_{1},T_{2},\ldots,T_{m}$ of $T$ such that $\sigma(T_{i})=0$ and $|T_{i}|=|G|$ for

$i\in[1,m].$ This implies that ${\mathsf{E}}_{m}(G)>{\mathsf{d}}(G)+m|G|-1.$ Hence,

[TABLE]

On the other hand, if $S$ is any sequence over $G$ such that $|S|\geq{\mathsf{E}}(G)+(m-1)|G|,$ then one can sequentially extract at least $m$ disjoint subsequences $S_{1},\ldots,S_{m}$ of $S,$ such that $\sigma(S_{i})=0$ in $G$ and $|S_{i}|=|G|.$ Thus,

[TABLE]

∎

Corollary 8.3**.**

If $m\to\infty,$ then ${\mathsf{E}}_{m}(G)\sim m|G|.$

Proof.

If $m\to\infty,$ then $\tfrac{{\mathsf{E}}_{m}(G}{m}\to|G|$ by the inequality (39). ∎

We now give an application of theorem 8.2. The formula ${\mathsf{E}}(G)={\mathsf{d}}(G)+|G|$ was proved for all finite Abelian groups and for some classes of finite non-Abelian groups (see equation (3) and [4, 9, 10, 32]). Thus

[TABLE]

holds for finite groups in the following classes: Abelian groups, nilpotent groups, groups in the form $C_{m}\ltimes_{\varphi}C_{mn},$ where $m,n\in\mathbb{N},$ dihedral and dicyclic groups and all non-Abelian groups of order $pq$ with $p$ and $q$ prime. Therefore, the the following conjecture can be proposed:

Conjecture 8.4**.**

(See [4, Conjecture 2] For any finite group $G,$ we have equation

[TABLE]