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

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 case) let E m ( G ) ( or η m ( G )) 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 ≤ exp( G ) , respectively). In this paper, we prove that if G is an Abelian group, then E m ( G ) = D ( G ) – 1 + m | G | , which generalizes Gao’s relation. Moreover, we examine the asymptotic behaviour of the sequences ( E m ( G )) m ≥1 and ( η m ( G )) m ≥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 end, we apply the Davenport constant to smooth numbers and make a natural conjecture in the non-Abelian case.


Introduction
We will define and investigate some generalizations of the Davenport constant, see (Alon & Dubiner, 1995;Edel et al., 2007;Freeze & Schmid, 2010;Gao & Geroldinger, 2006;Geroldinger & Halter-Koch, 2006;Olson, 1969aOlson, , 1969b;;Reiher, 2007;Rogers, 1963).Davenport's constant is connected with algebraic number theory as follows.For an algebraic number field K, let  K be its ring of integers and G the ideal class group of  K .Let x∈ K be an irreducible element.If  K is a Dedekind domain, then x P K i i r  1 x P K i i r  1 P i , where P i are prime ideals in  K (not necessarily distinct).The Davenport constant D(G) is the maximal number of prime ideals P i (counted with multiplicities) in the prime ideal decomposition of the integral ideal x K , see (Halter-Koch, 1992;Olson, 1969a).
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 (Girard, 2018).

General notation
Let ℕ denote the set of positive integers (natural numbers).
We set [a, b] = {x: a ≤ x ≤ b, x∈ℤ}, where a, b∈ℤ.Our notation and terminology is consistent with (Geroldinger & Ruzsa, 2009).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 ⊕C n 2 ⊕…⊕C n r with natural numbers 1<n 1 |n 2 |…|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 . We write any finite sequence S of l elements of G in the form (this is a formal Abelian product), where l is the length of S denoted by |S|, and ν g (S) is the multiplicity of g in S. S corresponds to the sequence (in the traditional sense) (g 1 , g 2 , …, g l ), where we forget the ordering of the terms.By σ(S) we denote the sum of ( ) ( ) v S g G : . The Davenport constant D(G) is defined as the smallest t∈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: The small Davenport constant d(G) is the maximal length of a zero-sum free sequence over |G|.If |G| is a finite Abelian group then d(G) = D(G)-1, see (Geroldinger & Ruzsa, 2009 [Definition 2.1.1.]).Alford, Granville and Pomerance (1994) used the bound (1) to prove the existence of infinitely many Carmichael numbers.Dimitrov (2007) used the Alon Dubiner constant (Alon & Dubiner, 1995) to prove the inequality: , 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, 1969a(Olson, , 1969b)).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 https://doi.org/10.37705/TechTrans/e2020027lower bound, see (Geroldinger & Schneider, 1992).The following Remark 2.1 lists some basic facts for the Davenport constant, see (Delorme, Ordaz & Quiroz, 2001;Geroldinger & Schneider, 1992;Schmid, 2011;Sheikh, 2017).
Remark 2.1.Let G be a finite additive Abelian group.
One can derive, for example, the following inequalities. ( We note that a sequence S over G of length |S| ≥ mD(G) can be partitioned into m disjoint subsequences S i of length |S i | ≥ D(G).Thus, each S i contains a non-empty zero-sum subsequence.Hence D m (G) ≤ mD(G).See also (Halter--Koch, 1992 [Proposition 1 (ii)]).

The m-wise zero-sum constant and the m-wise Erdös--Ginzburg-Ziv constant of G
In 1996, Gao and Caro independently proved that for any finite Abelian group, see (Caro, 1996;Gao, 1995Gao, , 1996)).For a proof in modern language we refer to (Geroldinger & Halter-Koch, 2006 [Proposition 5.7.9]); see also (Delorme, Ordaz & Quiroz, 2001;Gao, 1994;Hamidoune, 1996).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 E(G) and E m (G).

∎
We recall that by s (m) (G) we denote the smallest t∈ℕ∪{∞} such that every sequence S over G of length t contains at least m disjoint non-empty Proof.The proof runs along the same lines as the proof of Theorem 4.1.

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' with n points whose centroid is also a lattice point.This conjecture was proven by Christian Reiher (2007).In order to prove the generalization of this theorem, we will use 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 , …, S m ⊆ 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 (Edel et al., 2007; Harborth,   1973), 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.It is known that s(C n 2 ) = 4n-3 for all n>1, see (Girard & Schmid, 2019, [Theorem 2.3]).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 has m pairwise disjoint subsets S 1 , S 2 , …, S m each of cardinality n, the centroids of which are also lattice points.Finally, in the case of 2-dimensional Euclidean space, it is sufficient to use equation ( 7), from which we get s (m) (C n 2 ) = (m+3)n-3.The last equality completes the proof of Theorem 5.1.

∎
Remark 5.2.As we can see in Definition 3.1, Theorem 5.1 is also true when we replace sets with multisets.

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 ≤k (G),D m (G),η(G) constants.Our main result of this section is Theorem 6.11.Olson (1969b)  ) = (n+1)p-n (see (Olson, 1969a) or Remark 2.1,(a)) there exists a zero-sum subsequence Rearranging subscripts, we may assume that g 1 + g 2 + … + g ep = 0, where e∈ [1, n].The thesis is achieved if e∈ [1, n-1].If e = n we obtain a zero-sum sequence S = g 1 .g 2 .…. g np .Zero-sum sequence S contains a proper zero-sum subsequence S', since D(C p n ) = np-(n-1), and thus a zero-sum subsequence of length not exceeding ( )
Proof.We use Lemma 6.3 and Remark 3.3.

∎
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 be a subgroup of G. Then: We obtain that ' ∏ is a non-empty zero-sum subsequence of S.  ( ) By ( 11) (with and Corollary 6.2, we get ( ) ( )
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: Assume inductively that theorem is true for Q i.e.
We shall prove that there exists a subsequence of S with a length smaller than or equal to 2 Ω(h) l and a zero sum.Let b i = g i +Q∈G/Q.We consider the sequence Therefore, by Corollary 6.2 there exists at least j 0 pairwise disjoint sets . Thus, we obtain a zero-sum https://doi.org/10.37705/TechTrans/e2020027subsequence of S in G of length not exceeding Proof.We proceed by induction on n and h n .
Suppose that the inequality (22) holds for fixed n ≥ 3 and fixed h, such that h 1 >h≥1: Let p be a prime divisor of h 1 .Let H i * be a subgroup of index p of a group H i .Put . By inductive assumption, the inequality ( 22) holds for Q: We put s = (n-1) Ω(h n ) (2(h n -1)+(h n-1 -1)+…+(h 1 -1)+1) and let S = g 1 .g 2 .…. g s be a sequence of G.
We shall prove that there exists a subsequence of S with a length smaller than or equal to (n-1) Ω(h n ) h n and a zero sum.
Therefore, by Lemma 6.1 there exists at least j 0 pairwise disjoint sets such that sequence https://doi.org/10.37705/TechTrans/e2020027 In another words ( ) . By induction assumption ( 22) . Thus, we obtain a zero sum subsequence of S in G of length not exceeding ∎

The smooth numbers
First, we recall the notation of a smooth number.Let F={q 1 , q 2 , …, q r } be a subset of positive integers.A positive integer k is said to be smooth over a set F if k q q q = ⋅ ⋅ ⋅  , where e i are non-negative integers.Remark 7.1.Let n∈ℕ.Each smooth number over a set {q 1 n , q 2 n , …, q r n } is an n-th power of a suitable smooth number over the set {q 1 , q 2 , …, q r }.
Definition 7.2.Let {p 1 , p 2 , …, p r } be a set of distinct prime numbers.By c(n 1 , n 2 , …, n r ), we denote the smallest t∈ℕ such that every sequence M of smooth numbers over a set {p 1 , p 2 , …, 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 { } In the next theorem, we use notation , these two structures are isomorphic to one another.Theorem 7.3.Let n 1 , n 2 , …, n r be integers such that 1<n 1 |n 2 |…|n r .Then: Proof.It follows on the same lines as the proof of (Chintamani et al., 2012 [Theorem 1.6.]).First, we will prove that We put l=D(ℤ n 1 ⊕ ℤ n 2 ⊕ … ⊕ ℤ n r ).Let M=(m 1 , m 2 , …, m l ) be a sequence of smooth numbers with respect to F={p 1 , p 2 , …, p r }.
For all i∈[1, l], we have , where e i,j are non-negative integers.
We associate each m i with a i ∈ℤ n 1 ⊕ ℤ n 2 ⊕ … ⊕ ℤ n r under the homomorphism: where i∈ [1, l].
Thus, we get a sequence S = a 1 a 2 ⋅…⋅a l of elements of the group Therefore, there exists a non-empty zero sum subsequence T of S in ℤ n 1 ⊕ ℤ n 2 ⊕ … ⊕ ℤ n r .Let T=a j 1 a j 2 ⋅…⋅a j t , where ( ) where k∈ [1, r]. https://doi.org/10.37705/TechTrans/e2020027 Consider the subsequence N of M corresponding to T. We have N = (m j 1 , m j 2 , …, m j t ) and by equation ( 35), we get ( ) for some integers l k ≥ 0. Thus, the product For some integers l k ≥ 0. The subsequence T of S corresponding to N will sum up to the identity in ℤ n 1 ⊕ ℤ n 2 ⊕ … ⊕ ℤ n r .Therefore, (37) holds and we obtain (31).∎

Future work and the non-Abelian case
The constant E(G) has received a lot of attention over the last ten years.
Let G be any additive finite group.Let S = (a 1 , a 2 , …, a n ) be a sequence over G.We say that the sequence S is a zero-sum sequence if there exists a permutation σ:[1, n]⟶ [1, n] such that 0 = a σ(1) +…+a σ(n) .For a subset I ⊆ ℕ, let s I (G) denote the smallest t∈ℕ∪{0, ∞} such that every sequence S over G of length |S| ≥ t has a zero-sum subsequence S' of length |S'|∈I.The constants D(G) ≔ s ℕ (G) and E(G) ≔ s {|G|} (G) are classical invariants in zero-sum theory (independently of whether G is Abelian or not).We recall that for a given finite group G, we denote by d(G), the maximal length of a zero-sum free sequence over G.We call d(G) the small Davenport constant.We observe that the sequence T does not contain m disjoint non-empty subsequences T 1 ,T 2 ,… ,T m of T such that σ(T i ) = 0 and |T i | = |G| for i∈ [1, m].This implies that E m (G) > d(G)+m|G|-1.Hence E m (G) ≥ d(G)+m|G|.

∎
We now give an application of Theorem 8.2.The formula E(G) = d(G)+|G| was proved for all finite Abelian groups and for some classes of finite non-Abelian groups (see equation ( 3) and (Bass, 2007;Han, 2015;Han & Zhang, 2019;Oh & Zhong, 2019)).Thus E m (G) = d(G)+m|G| holds for finite groups in the following classes: Abelian groups, nilpotent groups, groups in the form C m ⋉ φ C mn , where m, n∈ℕ, dihedral and dicyclic groups and all non-Abelian groups of order pq with p and q prime.Therefore, the following conjecture can be proposed:

Conclusion
This paper makes a contribution to the theory of additive combinatorics.It provides an overview of the state of knowledge of the zero-sum problems and can be considered as an introduction to this theory.We have proven that if G is an Abelian group, then E m (G) = d(G)+m|G|.We have studied the asymptotic behaviour of the sequences (E m (G)) m≥1 and (η m (G)) m≥1 .For a prime p and a natural n ≥ 2, we have derived the inequality s ≤(n-1)p (C p n ) ≤ (n+1)p-n.We have proven a generalization of Kemnitz's conjecture.We have applied the Davenport constant to smooth numbers.Finally, we have shown some results in the non-Abelian case.

Theorem 4. 1 .
If G is a finite Abelian group of order |G|, then

)
Proof.The inequality D(H)+D(G/H)-1 ≤ D(G) is proven in(Halter-Koch, 1992  [Proposition 3 (i)]).Now we prove the inequality D(G) ≤ D D(H) (G/H) on the same lines as in(Delorme, Ordaz & Quiroz, 2001), we include the proof for the sake of completeness.If |S|≥D D(H) (G/H) isany sequence over G, then one can, by definition, extract at least D(H) disjoint non-empty subsequences S 1 , …, S D(H) |S such that σ(S i )∈H.Since over H of length D(H), there thus exists a non-empty subset I⊆[1, D(H)] such that ( ) sum in G/Q.In other words( )
in each of the following cases: ⊕C p ⊕C p n m , with p a prime number, n ≥ 2 and m a natural number coprime with p n (more generally, if ▶ G is a p-group; ▶ G has rank r ≤ 2; ▶ G = C p

Definition 3.1. Let
G be a finite Abelian group, and m, k be positive integers such that k ≥ exp(G), and ∅≠I ⊆ ℕ. 1.By s I (G) we denote the smallest t∈ℕ∪{∞} such that every sequence S (with repetition allowed) over G of length t contains a non-empty subsequence S' such that σ(S') = 0, |S'|∈I.We use notation s ≤k (G) or D k (G) to denote s I (G) if we denote the smallest t∈ℕ∪{∞} such that every sequence S over G of length t contains at least m disjoint non-empty subsequences S 1 , S 2 , …, S m such that σ(S i ) = 0, |S i |∈I. 3. Let E(G) ≔ s {|G|} (G), i.e. the smallest t∈ℕ∪{∞} such that every sequence S over G with length t contains non-empty subsequence S' such that σ(S') = 0, |S'| = |G|.Note that E(G) is the classical zero-sum constant.4. Let E m (G) ≔ s {|G|},m (G), i.e. the smallest t∈ℕ∪{∞} such that every sequence S over G with length t contains at least m non-empty subsequences S 1 , S 2 , …, S m such that σ( If p is a prime and G=C p e 1 ⊕…⊕C p e k is a p-group, then for a natural m 1)|G|.∎ https://doi.org/10.37705/TechTrans/e2020027Corollary 4.2.For every finite Abelian group, the sequence (E m (G)) m≥1 is an arithmetic progression with difference |G|.Corollary 4.3.
calculated s ≤p (C p 2 ) for a prime number p.No precise result is known for s ≤p(C p n ), where n≥3.We need two technical lemmas: Lemma 6.1.Let p be a prime number and n≥2.Then:s ≤(n-1)p (C p n ) ≤ (n+1)p-n.
the terms of N is = (m 1 , m 2 , …, m l ) of integers is a sequence of smooth numbers over a set F.By definition of l = c(n 1 , n 2 , …, n r ), there exists a non-empty subsequence