Multi-wise and constrained fully weighted Davenport constants and interactions with coding theory
Luz Elimar Marchan, Oscar Ordaz, Irene Santos, Wolfgang Schmid
aa r X i v : . [ m a t h . N T ] M a y MULTI-WISE AND CONSTRAINED FULLY WEIGHTEDDAVENPORT CONSTANTS AND INTERACTIONS WITHCODING THEORY
LUZ E. MARCHAN, OSCAR ORDAZ, IRENE SANTOS, AND WOLFGANG A. SCHMID
Abstract.
We consider two families of weighted zero-sum constants for finiteabelian groups. For a finite abelian group ( G, +), a set of weights W ⊂ Z , andan integral parameter m , the m -wise Davenport constant with weights W isthe smallest integer n such that each sequence over G of length n has at least m disjoint zero-subsums with weights W . And, for an integral parameter d ,the d -constrained Davenport constant with weights W is the smallest n suchthat each sequence over G of length n has a zero-subsum with weights W ofsize at most d . First, we establish a link between these two types of constantsand several basic and general results on them. Then, for elementary p -groups,establishing a link between our constants and the parameters of linear codesas well as the cardinality of cap sets in certain projective spaces, we obtainvarious explicit results on the values of these constants. Introduction
For a finite abelian group ( G, +) the Davenport constant of G is the smallest n such that each sequence g . . . g n over G has a non-empty subsequence the sum ofwhose terms is 0. This is a classical example of a zero-sum constant over a finiteabelian group, and numerous related invariants have been studied in the literature.We refer to [16] for a survey of the subject.An example are the multi-wise Davenport constants: the m -wise Davenportconstant, for some integral parameter m , is defined like the Davenport constantyet instead of asking for one non-empty subsequence with sum 0 one asks for m disjoint non-empty subsequences with sum 0. These constants were first consideredby Halter-Koch [23] due to their relevance in a quantitative problem of non-uniquefactorization theory. Delorme, Quiroz and the second author [11] high-lighted therelevance of these constants when using the inductive method to determine theDavenport constant itself.Another variant are constants defined in the same way as the Davenport con-stant, yet imposing a constraint on the length of the subsequence whose sum is0. A classical example are generalizations of the well-known Erd˝os–Ginzburg–Zivtheorem, where one seeks zero-sum subsequences of lengths equal to the exponentof the group or also equal to the order of the group. Another common constraintis to impose an upper bound on the length, for example again the exponent of Mathematics Subject Classification.
Key words and phrases. finite abelian group, weighted subsum, zero-sum problem, Davenportconstant, linear intersecting code, cap set.The research of O. Ordaz is supported by the Postgrado de la Facultad de Ciencias de laU.C.V., the CDCH project, and the Banco Central de Venezuela; the one of W.A. Schmid by theANR project Caesar, project number ANR-12-BS01-0011. the group, yielding the η -invariant of the group. Here, we refer to these constantsas constrained Davenport constants, more specifically the d -constrained Davenportconstant is the constant that arises when one asks for the existence of a non-emptyzero-sum subsequence of length at most d . There are numerous contributions tothis problem and we refer to [16, Section 6] for an overview.In addition to these classical zero-sum constants, in recent years there was con-siderable interest in weighted versions of these constants. There are several waysto introduce weights in such problems. One that received a lot of interest latelyis due to Adhikari et al. (see [1, 2, 31, 32] for some contributions, and [33] fora more general notion of weights) where for a given set of weights W ⊂ Z oneasks for the smallest n such that a sequence g . . . g n over G has a W -weightedsubsum that equals 0, that is there exists a subsequence g i . . . g i k and w j ∈ W such that P kj =1 w j g i j = 0, yielding the W -weighted Davenport constant of G .And, analogously, one defines the m -wise W -weighted Davenport constant and the d -constrained W -weighted Davenport (see Definition 3.1 for a more formal defini-tion). We refer to [22] for an overview on weighted zero-sum problems, includinga more general notion of weights, and to [24] for an arithmetical application of aweighted Davenport constant.In the present paper we investigate multi-wise weighted Davenport constantsand constrained weighted Davenport constants. We obtain some results for generalsets of weights W and general finite abelian groups, but our focus is on elementary p -groups and on the case that the set of weights is “full,” that is it contains allintegers except multiples of the exponent of the groups, which is the largest set ofweights for which the problem is non-trivial (see Section 2 for details). This workbuilds on and generalizes earlier work on the classical versions of these constantsby Cohen and Z´emor [9], Freeze and the last author [15], and Plagne and the lastauthor [28] for elementary 2-groups; in particular we establish that the problemscan be linked to problems in coding theory, as was known in the case of elementary2-groups.Moreover, for the case of elementary 3-groups the problem coincides with theplus-minus weighted problem, that is the problem for sets of weights { +1 , − } . Re-cently, constrained Davenport constants were investigated in this case by Godhino,Lemos, and Marques [19], and we improve several of their results.The organization of the paper is as follows. After recalling some standard termi-nology, we recall in Section 3 the definitions of the key invariants for this paper, andprove some general results on them. In particular, we show that for arbitrary setsof weights the multi-wise weighted Davenport constants are eventually arithmeticprogressions. This generalizes a result of Freeze and the last author [15] for theclassical case.We then focus on the fully-weighted case. First, in Section 4, we explain thelink to coding theory in a general way. The crucial difference to earlier workssuch as [9, 20, 28] is that here we are not restricted to binary linear codes andelementary 2-groups, but more generally can consider p -ary linear codes for someprime p and elementary p -groups (we deviate from the more usual convention totalk about q -ary codes, since on the one hand we only consider primes not primepowers and on the other hand to stay in line with the common usage for groups).We use this link in two different ways. In Section 5 we use known results on theoptimal minimal distance of codes of small dimension and length to obtain the ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 3 exact values of or good bounds for the constrained Davenport constants of someelementary p -groups for small exponent and rank. In that section, among otherresults, we use the fact that linear codes of minimal distance four are closely linkedto cap sets in projective spaces. In particular, we obtain a characterization of the3-constrained fully-weighted Davenport constants of elementary p -groups in termsof the maximal cardinality of cap sets in projective spaces over the fields with p elements. For context, we recall that a link between a certain classical zero-sumproblem and the cardinality of caps in affine spaces was known (see [12]). In theother direction, in Section 7, we use asymptotic bounds on the parameters of codesto obtain asymptotic bounds on our constants. In that section we also obtainsome lower bounds for our constants that can be interpreted as existence-proofsfor a certain type of codes related to the notion of intersecting codes introducedby Cohen and Lempel [8]. Moreover, in Section 6 we determine all the fully-weighted multi-wise Davenport constants for elementary p -groups of rank at most2, and for C (for the values of the multi-wise Davenport constant in the classicalcase for these groups see [23] and [4], respectively). The last example shows aninteresting additional phenomenon and illustrates the difficulty of obtaining moregeneral results. 2. Preliminaries
We recall some standard terminology and notation. We denote by N the positiveintegers and we set N = N ∪ { } . We denote by log the natural logarithm andby log b the logarithm to the base b . For abelian groups, we use additive notationand we denote the neutral element by 0. For n ∈ N let C n denote a cyclic groupof order n . For p a prime number we denote by F p the field with p elements. Foreach finite abelian group there exist uniquely determined 1 < n | · · · | n r such that G ∼ = C n ⊕ · · · ⊕ C n r . One calls n r the exponent of G , and it is denoted by exp( G );moreover r is called the rank of G . We say that G is a p -group if its exponent is aprime power and we say that G is an elementary p -group if the exponent is a prime(except for the trivial group).We recall that a (finite) abelian group is a Z -module, and also a Z / exp( G ) Z module. In particular, an elementary p -group is a vector space over the field withcardinality exp( G ) (except for the trivial group).A family ( e , . . . , e k ) of non-zero elements of a finite abelian group is calledindependent if P ki =1 a i e i = 0 with a i ∈ Z implies that a i e i = 0 for each i . For anelementary p -group, a family of non-zero elements is independent in this sense ifand only if it is linearly independent when considering the group as vector space inthe way given above. We call an independent generating set of non-zero elementsa basis; in the case of elementary p -groups this notion of basis coincides with theusual one for vector spaces.A key notion of this paper are sequences. We recall some terminology andnotation. A sequence over G , a subset of a finite abelian group G , is an elementof F ( G ), the free abelian monoid over G . We use multiplicative notation. Thismeans that a sequence S can be written uniquely as Q g ∈ G g v g with v g ∈ N ,or uniquely except for ordering as S = g . . . g s where g i ∈ G and repetitionof elements can occur. We denote by | S | = s the length of the sequence, by σ ( S ) = P si =1 g i the sum of S , and by v g ( S ) = v g the multiplicity of g in S .Formally, a subsequence of S is a divisor T of S in F ( G ), that is T = Q i ∈ I g i for LUZ E. MARCHAN, OSCAR ORDAZ, IRENE SANTOS, AND WOLFGANG A. SCHMID some I ⊂ { , . . . , s } , which thus matches the notion of subsequence used in othercontexts.We call subsequences T , . . . , T t of S disjoint if Q tj =1 T j | S . This can be ex-pressed as saying that there are pairwise disjoint subsets I j ⊂ { , . . . , s } , for j ∈ { , . . . , t } , such that T j = Q i ∈ I j g i . However, it should be noted that dis-joint subsequences of a sequence can have elements in common; this can happen incase they appear with multiplicity greater than one in the original sequence.A sequence is called squarefree if no element appears with multiplicity greaterthan 1. These sequences could be identified with sets; however, it is sometimesadvantageous to keep the notions separate.A sequence over G is called a zero-sum sequence if its sum is 0 ∈ G , and azero-sum sequence is called a minimal zero-sum sequence if it is non-empty anddoes not have a proper and non-empty subsequence that is a zero-sum sequence.We denote the set of all minimal zero-sum sequences over G by A ( G ).For a subset W ⊂ Z a W -weighted sum of S is an element of the form P si =1 w i g i with w i ∈ W . We denote by σ W ( S ) the set of all W -weighted sums of S . Thisnotion extends the notion of sum in the classical case; indeed, σ { } ( S ) = { σ ( S ) } . A W -weighted subsum of S is a W -weighted sum of a non-empty subsequence of S ; wechoose to exclude the empty sequence, since this is more convenient in general (inthe rare cases we admit the empty sequence we mention it locally). The length ofa subsum is just the length of the respective subsequence. A W -weighted zero-sum(or zero-subsum) is merely a W -weighted sum (or subsum) whose value is 0 ∈ G .Of course, the W -weighted sums of a sequence S depend only on the image of W under the standard map to Z / exp( G ) Z . We thus could restrict to consideringsets of weights contained in { , , . . . , exp( G ) − } . The problems we study becometrivial if W contains 0 or more generally a multiple of exp( G ). Thus, we call a setof weights W trivial with respect to exp( G ) if it contains a multiple of exp( G ).As mentioned in the introduction in later sections we focus on the problem forthe “full” set of weights Z \ exp( G ) Z or equivalently { , . . . , exp( G ) − } that isthe largest set of weights that is not trivial; we reserve the letter A for this set ofweights. Since in these investigations at least the exponent of the groups underconsideration will always be clear and essentially fixed, no confusion should arisefrom the fact that the set A depends implicitly on the exponent of the group.Moreover, we call a set of weights W multiplicatively closed modulo exp( G ) ifthe image of W under the standard map from Z to Z / exp( G ) Z is multiplicativelyclosed. 3. Main definitions and general results
We begin by stating in a more formal and in part more general way the definitionof the weighted versions of the constrained and multi-wise Davenport constants. Fora discussion of earlier appearances of the m -wise Davenport constant we refer tothe introduction; in [17] a definition of s W,L ( G ) without weights was given. Definition 3.1.
Let G be a finite abelian group. Let W ⊂ Z be a non-emptysubset.(1) For a non-empty set L ⊂ N , the L -constrained W -weighted Davenportconstant of G , denoted s W,L ( G ), is the smallest n ∈ N ∪ {∞} such thateach S ∈ F ( G ) with | S | ≥ n has a W -weighted zero-subsum of length in ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 5 L . For the special case that L = { , . . . , d } , we denote the constant by s W, ≤ d ( G ) and call it the d -constrained W -weighted Davenport constant.(2) For m ∈ N , the m -wise W -weighted Davenport constant of G , denoted D W,m ( G ), is the smallest n ∈ N ∪ {∞} such that each S ∈ F ( G ) with | S | ≥ n has at least m disjoint W -weighted non-empty zero-subsums.It is easy to see that D W,m ( G ) is in fact always finite; for a characterization ofthe finiteness of s W, ≤ d ( G ) see Lemma 3.4.Sometimes we use simplified versions of this notation for common special cases.When no set of weights is indicated, we mean the set of weights W = { } . Moreover, D W ( G ) = D W, ( G ). For some further conventions see the discussion before Lemma3.5.We make some simple observations how these constants depend on the parame-ters; we continue to use the just introduce notations. Furthermore, let H ⊂ G bea subgroup, let W ′ ⊂ W , L ′ ⊂ L , and let m ′ ≤ m a positive integer. Then • D W,m ( H ) ≤ D W,m ( G ) and s W,L ( H ) ≤ s W,L ( G ). • D W,m ( G ) ≤ D W ′ ,m ( G ) and s W,L ( G ) ≤ s W ′ ,L ( G ). • D W,m ′ ( G ) ≤ D W,m ( G ) and s W,L ( G ) ≤ s W,L ′ ( G ).Some of the results below allow to refine the inequalities given above. Thefollowing results show that multi-wise weighted Davenport constants are eventuallyarithmetic progressions. In the classical case it is known that they are eventuallyarithmetic progressions with difference equal to the exponent of the group (see [15,Lemma 5.1]); in the presence of weights the difference depends on the set of weightsand the exponent of the group, in a form we make precise below. We mention that asomewhat similar phenomenon occurs in recent investigations of a quantity relatedto the m -wise Davenport constant for non-commutative finite groups [10]. Theorem 3.2.
Let G be a finite abelian group. Let W ⊂ Z be a set of weights,and let W denote its image under the standard map from Z to Z / exp( G ) Z . Then ( D W,m ( G )) m ∈ N is eventually an arithmetic progression with difference min {| U | : U ∈A ( W ) } . We point out that the proof of this result is not constructive, in the sense thatit does not yield an actual upper bound on an M such that ( D W,m ( G )) m ≥ M isan arithmetic progression. This could likely be overcome in a similar way as inthe classical case (see [15, Proposition 6.2]). However, the argument should besomewhat lengthy and the bound very weak. Moreover, in Corollary 3.13 we seethat in many cases of interest a not too bad bound can be obtained in another way.Thus, we do not pursue the question of obtaining a general explicit bound.We split the proof of the result into several lemmas that are also useful in theirown right. In all the results below let G be a finite abelian group, let W ⊂ Z be aset of weights, and let W denote its image under the standard map to Z / exp( G ) Z .Furthermore, let e W ( G ) = min {| U | : U ∈ A ( W ) } . This notation is chosen for thisspecific context only, to stress the role min {| U | : U ∈ A ( W ) } plays in our context;it is a parameter to describe W -weighted zero-sum constants of G . The quantitymin {| U | : U ∈ A ( H ) } for H a subset of a finite abelian group H also comes up inother contexts (see below).We make an additional definition that allows to phrase some results on the multi-wise Davenport constants in a concise way; the definition makes sense in view ofthe result above. LUZ E. MARCHAN, OSCAR ORDAZ, IRENE SANTOS, AND WOLFGANG A. SCHMID
Definition 3.3.
Let G , W , W , and e W ( G ) as above. Then, let D W, ( G ) denotethe integer such that D W,m ( G ) = D W, ( G ) + m e W ( G ) for all sufficiently large m .Moreover, let m W ( G ) denote the minimal integer such that D W,m ( G ) = D W, ( G ) + m e W ( G ) for each m ≥ m W ( G ).We start by collecting some simple remarks on e W ( G ). We have e W ( G ) ≤ exp( G ). This follows from the fact that the Davenport constant of Z / exp( G ) Z isexp( G ). Equality holds if and only if modulo exp( G ) the set W contains only asingle element that generates Z / exp( G ) Z , that is, in the classical case. Moreover,in the plus-minus weighted case, that is W = { +1 , − } , as well as in the fullyweighted case, that is W = { , . . . , exp( G ) − } , we have e W ( G ) = 2. Of course,there is a large variety of other possible values for e W ( G ) in general, for example e { , } ( G ) = ⌈ exp( G ) / ⌉ . Indeed, we recall that the problem of determining theminimal cross number, a notion similar to that of length, of minimal zero-sumsequences plays an important role in investigations on the elasticity (see [6]); forelementary p -groups the problems in fact coincide (up to a scaling constant) andwe refer to [7] for recent investigations related to this problem.Now, we investigate under which conditions on d we have that s W, ≤ d ( G ) is fi-nite; the bound we give in case it is finite is rather crude, and mainly given fordefiniteness. The actual value is investigated in latter sections in certain cases. Lemma 3.4. (1)
We have s W, ≤ d ( G ) = ∞ for d < e W ( G ) and s W, ≤ d ( G ) ≤ ( e W ( G ) − | G | + 1 for d ≥ e W ( G ) . (2) We have s W, ≤ D W ( G ) ( G ) = D W ( G ) .Proof.
1. Let g ∈ G an element of order exp( G ). We assert that for every ℓ ∈ N the sequence g ℓ does not have a nonempty W -weighted zero-subsum of lengthstrictly less than e W ( G ). We note that P ri =1 w i g with w i ∈ W is 0 if and only if P ri =1 w i ≡ G )). In other words, w . . . w r is a zero-sum sequence over Z / exp( G ) Z . Thus, the minimal length of a nonempty W -weighted zero-sum onlyinvolving the group element g is e W ( G ), establishing our claim. Thus, s W, ≤ d ( G ) = ∞ for d < e W ( G ).Now, let S be a sequence over G of length at least ( e W ( G ) − | G | +1. It containsa subsequence of the form h e W ( G ) for some h ∈ G . Let w , . . . , w e W ( G ) such thattheir sum is 0 modulo exp( G ); the existence is guaranteed by the definition of e W ( G ). Then, P e W ( G ) i =1 w i h is a W -weighted zero-subsum of h exp( G ) and thus of S ;note that the order of h might not be equal to exp( G ), yet it is always a divisor ofexp( G ) and this suffices. Thus, s W, ≤ d ( G ) ≤ ( e W ( G ) − | G | + 1 for d ≥ e W ( G ).2. Every sequence of length D W ( G ) has a non-empty W -weighted zero-subsum,which of course has length at most D W ( G ) and thus D W ( G ) ≤ s W, ≤ D W ( G ) ( G ). Theconverse inequality is obvious. (cid:3) We point out that in the classical case, that is W = { } , of course e W ( G ) =exp( G ). The fact that this is the first value of d for which the constant is finite inthat case can be seen as one reason that there is a particular focus on s W, ≤ e W ( G ) ( G ),typically denoted η W ( G ) and also s W, { exp( G ) } ( G ) typically denoted just s W ( G ). Lemma 3.5.
Let d, ℓ ∈ N with d ≥ and ℓ ≤ s W, ≤ d − ( G ) . Then D W, ⌈ ℓ/d ⌉ ( G ) ≥ ℓ .In particular, for each m ∈ N we have D W,m ( G ) ≥ m e W ( G ) . ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 7
Proof.
Let S be a sequence of length ℓ −
1. Since | S | < s W, ≤ d − ( G ), it followsthat each W -weighted zero-subsum of S has length at least d . Since ⌈ ℓ/d ⌉ d ≥ ℓ > | S | , it follows that S cannot have ⌈ ℓ/d ⌉ disjoint W -weighted zero-subsums. Thus, D ⌈ ℓ/d ⌉ ( G ) > | S | , establishing the first part of the result.To see the second part, we recall from Lemma 3.4 that s W, ≤ e W ( G ) − ( G ) = ∞ .Thus, for each m ∈ N , we have m e W ( G ) ≤ s W, ≤ e W ( G ) − ( G ) and thus by the firstpart with ℓ = m e W ( G ) and d = e W ( G ), the claim follows. (cid:3) We establish an upper-bound for D W,m +1 ( G ) involving D W,m ( G ) and s W, ≤ d ( G ).The result is similar to a result in [15, Proposition 3.1] for the classical case. How-ever, the weighted version is minimally weaker; in the classical version we can take s W, ≤ d ( G ) − s W, ≤ d ( G ). Lemma 3.6.
We have D W,m +1 ( G ) ≤ min d ∈ N max { D W,m ( G ) + d, s W, ≤ d ( G ) } .Proof. Let S be a sequence over G such that | S | ≥ min d ∈ N max { D W,m ( G ) + d, s W, ≤ d ( G ) } . Let d ∈ N such that the minimum is attained for d . Since | S | ≥ max { D W,m ( G ) + d , s W, ≤ d ( G ) } , it follows that S has a W -weighted zero-subsum of length at most d ; let T denote the corresponding subsequence of S . Then, | ST − | ≥ D W,m ( G ).By the very definition of D W,m ( G ) the sequence ST − has m disjoint W -weightedzero-subsums. Thus, we have established the existence of m + 1 disjoint W -weighted zero-subsums of S , showing that D W,m +1 ( G ) ≤ min d ∈ N max { D W,m ( G ) + d, s W, ≤ d ( G ) } . (cid:3) We now can prove Theorem 3.2.
Proof of Theorem 3.2.
For m ≥ ( e W ( G ) − | G | + 1 = m we have, using Lemma3.4 and Lemma 3.5,max { D W,m ( G ) + e W ( G ) , s W, ≤ e W ( G ) ( G ) } = D W,m ( G ) + e W ( G ) . Thus by Lemma 3.6 we have for m ≥ m that D W,m +1 ( G ) ≤ D W,m ( G ) + e W ( G )).Consequently the sequence ( D W,m ( G ) − m e W ( G )) m ≥ m is non-increasing. And,by Lemma 3.5 it is a sequence of non-negative integers. Therefore it is eventuallyconstant, establishing the result. (cid:3) We collect some lemmas that relate the values of the constants we investigatefor a group G to those of some subgroup H and the quotient group G/H . Theseresults generalize results known in the classical case and sometimes also in morerestricted cases with weights.We start with two lower bounds, which among others can be useful to assertin certain cases that the value of D W,m ( G ) or s W, ≤ d ( G ) are strictly greater thanthe respective values for a proper subgroup (in other words the extremal sequenceswith respect to these constants generate the group). These bounds generalize resultsknown in the classical case, see for example [18, Section 6.1]. Lemma 3.7.
Let m, m , m ∈ N such that m ≥ m + m − . Let H be a subgroupof G . Then D W,m ( G ) ≥ D W,m ( H ) + D W,m ( G/H ) − .Proof. Let S be a sequence over H of length D W,m ( H ) − m disjoint W -weighted zero-subsums. Let T be a sequence over G of length LUZ E. MARCHAN, OSCAR ORDAZ, IRENE SANTOS, AND WOLFGANG A. SCHMID D W,m ( G/H ) − T over G/H obtained from T by ap-plying the canonical epimorphism to each element in T does not have m disjoint W -weighted zero-subsums.We claim that ST does not have m disjoint W -weighted zero-subsums. We startby analyzing the types of W -weighted zero-subsums there are. Let R | ST suchthat 0 ∈ σ W ( R ). There are S ′ | S and T ′ | T such that R = S ′ T ′ . • If T ′ is empty, then obviously 0 ∈ σ W ( S ′ ), and this yields a W -weightedzero-subsum of S • If T ′ is non-empty, then, since σ W ( T ′ ) ∩ ( − σ W ( S ′ )) = ∅ and since σ W ( S ′ ) ⊂ H , we get that σ W ( T ′ ) contains an element from H . Thus, the image of T ′ under the canonical epimorphism gives rise to a W -weighted subsum over G/H .Suppose we have R . . . R v | ST with R i non-empty and 0 ∈ σ W ( R i ) for each i . Wewrite R i = S i T i such that S . . . S m | S and T . . . T m | T .For each i we know that 0 ∈ σ W ( S i ) where S i is non-empty or 0 ∈ σ W ( T i ) where T i is non-empty. Let us denote the set of indices i for which the former holds by I S and the complement of I S in { , . . . , m } by I T .Since Q i ∈ I S S i | S and S has at most m − W -weighted zero-subsumswet get that | I S | ≤ m −
1. Since Q i ∈ I T T i | T and T has at most m − W -weighted zero-subsums wet get that v = | I S | + | I T | ≤ m − m − < m .Thus, ST does not have m disjoint W -weighted zero-subsums and D W,m ( G ) − ≥| ST | ≥ ( D W,m ( H ) −
1) + ( D W,m ( G/H ) − (cid:3) Lemma 3.8.
Let d ∈ N . Let H be a subgroup of G , then s W, ≤ d ( G ) ≥ s W, ≤ d ( H ) + s W, ≤ d ( G/H ) − .Proof. Let T be a sequence over G whose image in G/H does not have a W -weightedzero-subsum. Then T does not have a W -weighted subsum that is an element of H . Thus, for each S a sequence over H that does not have a W -weighted zero-subsum, we have that ST does not have a W -weighted zero-subsum. It follows that s W, ≤ d ( G ) > ( s W, ≤ d ( H ) −
1) + ( s W, ≤ d ( G/H ) −
1) establishing the claim. (cid:3)
We observe that this lemma yields some information on the structure of the se-quences over G of length s W, ≤ d ( G ) − W -weighted subsum oflength at most d . Namely, it follows, denoting by H the subgroup generated by theelements in such a sequence, that s W, ≤ d ( G/H ) = 1. Under various circumstancesthis implies that
G/H is trivial, which means that the elements in such a sequencegenerate the group G . In particular, this is the case for G an elementary p -groupand W a non-trivial set of weights modulo p .The following two results establish ‘inductive’ upper bounds on our constantsin terms of the constants for a subgroup and the quotient group with respect tothis subgroup. The two preceding lemmas could be thought of as ‘inductive’ lowerbounds; this terminology is not so common. For an overview of the inductivemethod see [18, Section 5.7]. Our results expand known results to this more generalcontext, containing the classical ones as a special case; yet in fact Lemma 3.10 iseven more general than the existing results in the classical case. For these ‘inductive’results we need to impose some restriction on the sets of weights, namely that W is multiplicatively closed modulo exp( G ). ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 9
Before phrasing our actual results we explain the relevance of this condition andmore generally the inductive method. This method consists of splitting a zero-sumproblem for G into a problem for a subgroup H and a problem for the quotient group G/H . Let π : G → G/H denote the natural epimorphism. Let S be a sequence over G . To find a zero-sum subsequence of S one can proceed in the following way. Oneconsiders the sequence π ( S ) over G/H , obtained by applying π to each element in S , for simplicity of notation we denote π ( S ) by S and do alike for subsequences.Suppose we can assert the existence of disjoint zero-sum subsequences S . . . S k | S .The condition that S i is a zero-sum sequence in G/H means that the sum of S i is an element of H . Thus, denoting s i = σ ( S i ), we have a sequence s . . . s k over H . Now, if this sequence has a zero-sum subsequence Q i ∈ I s i , then it follows that Q i ∈ I S i is a zero-sum subsequence of S .We point out that we have used that we can take the sum of a sequence in aniterated way, namely we have σ ( σ ( S ) . . . σ ( S k )) = σ ( S . . . S k ). It is at this pointthat trying to directly carry over the results from the classical case to the problemwith weights would fail.We now phrase a generalization of this for the weighted case. Let s i ∈ σ W ( S i )for each i , then for W multiplicatively closed modulo exp( G ), we have(3.1) σ W ( s . . . s k ) ⊂ σ W ( S . . . S k ) . To see this just recall that an element of σ W ( s . . . s k ) is of the form P ki =1 w i s i with w i ∈ W while each s i is of the form P j ∈ J i v ij g j with v ij ∈ W where S i = Q j ∈ J i g j ;and, since modulo exp( G ) we have that w i v ij is again an element of W , we havethat P j ∈∪ ki =1 J i ( w i v ij ) g j is an element of σ W ( S . . . S k ).We now formulate our inductive bounds for the two types of constants that westudy. For the classical Davenport constant this result appears in [11, Proposition2.6] Lemma 3.9.
Let m ≥ and let H be a subgroup of G . Suppose W is multiplica-tively closed modulo exp( G ) . Then D W,m ( G ) ≤ D W, D W,m ( H ) ( G/H ) . Proof.
For notational simplicity let D = D W, D W,m ( H ) ( G/H ). Let S be a sequenceover G of length D . We need to show that S has m disjoint W -weighted zero-subsums. By the definition of D and considering S the sequence obtained from S by applying the canonical epimorphism from G to G/H to each element, it followsthat S has D W,m ( H ) disjoint W -weighted zero-subsums (in G/H ). Thus, S has D W,m ( H ) disjoint subsequences S i such that σ W ( S i ) contains an element from H ;denote this element by s i .The sequence T = s . . . s D W,m ( H ) is thus a sequence over H . By the definitionof D W,m ( H ), there exists T . . . T m | T with T j non-empty and 0 ∈ σ W ( T j ) for each j . Let I j ⊂ { , . . . , D W,m ( H ) } , for j ∈ { , . . . , m } , be disjoint subsets such that T j = Q i ∈ I j s i .Since W is multiplicatively closed, we have by (3.1) that σ W ( Y i ∈ I j s i ) ⊂ σ W ( Y i ∈ I j S i ) . Consequently, R j = Q i ∈ I j S i for j ∈ { , . . . , m } are non-empty disjoint subse-quences of S such that 0 ∈ σ W ( R j ). Thus, we established the existence of m pairwise disjoint W -weighted zero-subsums of S . (cid:3) Lemma 3.10.
Let L , L ⊂ N be non-empty subsets, and let H be a subgroup of G . Suppose W is multiplicatively closed modulo exp( G ) . Then s W,L ( G ) ≤ (max L )( s W,L ( H ) −
1) + s W,L ( G/H ) where L = ∪ l ∈ L lL and lL denotes the l -fold sumset of L .Proof. Let S be a sequence over G of length (max L )( s W,L ( H ) − s W,L ( G/H ) = t . Let S be the sequence obtained from S by applying the canonical epimorphismfrom G to G/H to each element. Since t ≥ s W,L ( G/H ) there exists a subsequence T | S with | T | ∈ L , and in particular | T | ≤ max L , such that 0 ∈ σ W ( T )that is there exists some element s ∈ H such that s ∈ σ W ( T ). We now consider S = T − S . If | S | ≥ s W,L ( G/H ) we get T | S with the analogous properties.Continuing in this way we get s W,L ( H ) = c disjoint subsequences T . . . T c | S with | T i | ∈ L and such that there exists some s i ∈ σ W ( T i ) ∩ H . By definition of c thesequence s . . . s c has a W -weighted subsum of length in L , say 0 ∈ σ W ( Q j ∈ J s j )with J ⊂ { , . . . , c } and | J | ∈ L . Now, by (3.1), since W is multiplicatively closedmodulo exp( G ), we have σ W ( Q j ∈ J s j ) ⊂ σ W ( Q j ∈ J T j ) and thus Q j ∈ J T j has 0 as a W -weighted sum. Since | T j | ∈ L for each j , we have that the length of Q j ∈ J T j isin | J | L , which is a subset of ∪ l ∈ L lL . Thus, we have found a W -weighted subsumof S whose length is in ∪ l ∈ L lL , establishing the result. (cid:3) We use the result above with special choices of L and L , to make explicit someconsequences of particular relevance to our investigations. The second part of thiscorollary generalizes [19, Proposition 6], an inductive result in the weighted case.For the classical case one can find various such results in the literature, especially for s ( G ) and η ( G ) they are well-known (see for example [16, Section 6]). For example,they allow in combination with results for p -groups, to determine the exact valueof s ( G ) and η ( G ) for groups of rank at most 2 (see [18, Theorem 5.8.3]). For somerecent results for groups of higher rank we refer to [14, 13, 29]. Corollary 3.11.
Let d , d ≥ and let H be a subgroup of G . Suppose W ismultiplicatively closed modulo exp( G ) . Then (1) s W, ≤ d d ( G ) ≤ d ( s W, ≤ d ( H ) −
1) + s W, ≤ d ( G/H ) . (2) s W, { d d } ( G ) ≤ d ( s W, { d } ( H ) −
1) + s W, { d } ( G/H ) .Proof. For 1., we apply Lemma 3.10 with L = { , . . . , d } and L = { , . . . , d } .Since in this case ∪ l ∈ L lL ⊂ { , . . . , d d } the result follows.In the same way, with L = { d } and L = { d } and thus L = ∪ l ∈ L lL = { d d } , we get 2. (cid:3) We end this section with a recursive lower bound for D W,m ( G ). At first thisbound could seem weak, but in view of Lemma 3.6 we note that it is in some senseoptimal in case e W ( G ) = 2, which covers various cases of interest (see the discussionafter Definition 3.3). Indeed, this bound allows us to obtain a more explicit versionof Theorem 3.2. We also note that D W,m +1 ( G ) ≥ D W,m ( G )+1 always holds; adding0 to a sequence increases the maximal number of disjoint W -weighted zero-subsumsby exactly one. ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 11
Lemma 3.12.
Let | G | > and let m ∈ N . Suppose that W is multiplicatively closedmodulo exp( G ) and non-trivial modulo exp( G ) . Then D W,m +1 ( G ) ≥ D W,m ( G ) + 2 .Proof. Let S be a sequence of length D W,m ( G ) − m disjoint W -weighted zero-subsums. Let g ∈ G with ord g = exp( G ). We consider S ( − g ) g .The result follows if we can show that S ( − g ) g does not have m + 1 disjoint W -weighted zero-subsums. Assume to the contrary there are T . . . T m +1 | S such that0 ∈ σ W ( T i ) for each i . We may assume that g | T and ( − g ) | T ; if only one ornone of the T i ’s would contain elements from ( − g ) g then the remaining sequenceswould be already subsequences of S , contradicting the assumption that S does notcontain m disjoint W -weighted subsums. Since W is non-trivial modulo exp( G )and ord g = exp( G ), it is clear that g − T and ( − g ) − T are non-empty.Thus, we have − wg ∈ σ W ( g − T ) and − w ′ ( − g ) ∈ σ W (( − g ) − T ). Conse-quently, w ′ ( − wg ) ∈ w ′ · σ W ( g − T ) = σ W ( g − T ), where for the last equality weused that W is multiplicatively closed modulo exp( G ), and likewise w ( − w ′ ( − g )) ∈ σ W (( − g ) − T ).Thus − ww ′ g ∈ σ W ( g − T ) and ww ′ g ∈ σ W (( − g ) − T ), and therefore 0 ∈ σ W ((( − g ) g ) − T T ). Yet, ((( − g ) g ) − T T ) T . . . T m +1 | S , and we have m disjoint W -weighted zero-subsums of S , a contradiction. (cid:3) For the case e W ( G ) = 2 and W multiplicatively closed modulo exp( G ), we nowobtain some explicit upper bound for m W ( G ), the index at which D W,m ( G ) startsto be an arithmetic progression. Corollary 3.13.
Suppose that W is multiplicatively closed modulo exp( G ) and that e W ( G ) = 2 . Then D W,m +1 ( G ) = D W,m ( G ) + 2 for each m ≥ | G | . In particular, m W ( G ) ≤ | G | .Proof. This is a direct consequence of Lemma 3.6 and Lemma 3.12. (cid:3)
We remark that the bound on m W ( G ) could be somewhat improved even withthe methods at hand; yet we merely meant to give some explicit bound here.4. Coding theory and weighted sequences
In this section, we develop the link between fully-weighted zero-sum problemsand problems on linear codes that was mentioned already in the introduction. Werecall that for elementary 2-groups, this link was already known; note that in factfor these groups the fully-weighted problem coincides with the classical one, andthe connection came up in that context (see [9, 28]). Furthermore, we recall thatfor elementary 3-groups the fully-weighted problem coincides with the plus-minusweighted problem, which is of particular interest.Before we discuss the link to coding theory we recall some basic facts relatedto fully-weighted zero-sum problems over elementary p -groups. Let p be a primenumber. Let G be a group with exponent p , and let A = { , . . . , p − } denote thefull set of weights. Recall that G can be considered in a natural way as a vectorspace over the field with p elements. We also identify the elements of A with thenon-zero elements of F p in the natural way. Furthermore, we recall that a sequence S = g . . . g n over G has no A -weighted zero-subsum if and only if ( g , . . . , g n ) islinearly independent, in particular(4.1) D A ( C rp ) = r + 1 and s A, ≤ r +1 ( C rp ) = r + 1 . Now, we briefly recall some notions from coding theory in a way suitable for ourapplication. As said above, C np is in a natural way a vector space over F p the fieldwith p elements. We implicitly fix some basis of C np , and thus can write its elementssimply as n -tuples of elements of F p : C np = { ( x , x , . . . , x n ) : x i ∈ F p , ≤ i ≤ n } . A p -ary linear code of length n and dimension k is a subspace C ⊂ C np of di-mension k . Briefly, we say that C is an [ n, k ] p -code. The elements of the code C are called codewords. The support of an element x = ( x , x , . . . , x n ) of C np , isthe subset of { , . . . , n } corresponding to the indices of non-zero coordinates x i .(This is the usage of the word ‘support’ common in coding theory; in the contextof zero-sum sequences ‘support’ typically has a different meaning.) The weight ofan element x ∈ C np , denoted by d ( x ), is the cardinality of the support of x . Theminimal distance of a code C , denoted d ( C ), is equal to the minimum d ( x ) with x anon-zero codeword of C ; that is, the minimal distance of the code C , is equal to theminimum cardinality of the support of a non-zero element of C . If C has minimal-distance d , then we say that C is a [ n, k, d ] p -code. Since in the present paper weonly consider linear codes, we choose the above quick, but not very intuitive way,to introduce the minimal distance.A parity check matrix of an [ n, k ] p -code C is a matrix H of dimension ( n − k ) × n (with full rank) over F p such that c ∈ C if and only if Hc = 0 (where we consider c as a column vector). For H = [ g | . . . | g n ], we can interpret the columns g i as elements of C n − kp , where again some basis is fixed, and in this way one canassign to an [ n, k ] p -code a sequence S = g . . . g n of length n over C n − kp . Now, c = ( c , . . . , c n ) ∈ C means that P ni =1 c i g i = 0. If we let I denote the support of c ,then we have P i ∈ I c i g i = 0 for non-zero c i .In other words P i ∈ I c i g i = 0 is an A -weighted zero-subsum of S (possibly theempty one); its length is exactly the cardinality of the support of c , that is d ( c ).Conversely, if P i ∈ J c ′ i g i = 0 is a (possibly empty) A -weighted zero-subsum of S ,then c ′ = ( c ′ , . . . , c ′ n ), where we set c ′ i = 0 for i / ∈ J , is an element of C withweight | J | , the length of the subsum. Of course, 0 ∈ C corresponds to the empty A -weighted zero-subsum. Thus, we have a direct correspondence between non-zerocodewords and A -weighted zero-subsums.We summarize these results in the lemmas below (recall A = { , . . . , p − } ). Lemma 4.1.
The minimal distance of a p -ary linear code C is equal to the minimallength of a A -weighted zero-subsum of columns of a parity check matrix of C .Proof. This is immediate by the discussion just above. (cid:3)
Lemma 4.2.
Let S = g . . . g n be a sequence over C rp (with some fixed basis) suchthat the set of all g i ’s is a generating set of C rp . Moreover, let H = [ g | . . . | g n ] denote the r × n matrix over the field F p (we identify the g i ’s with their coordinatevectors, in column-form). Then the code C S with parity check matrix H is an [ n, n − r ] p -code. And, each [ n, n − r ] p -code can be obtained in this way.Proof. Since the set of g i ’s is a generating set of C rp , the matrix H has full rank,i.e. rank r . Therefore, C S is an ( n − r )-dimensional subspace of C np . This givesthe first claim. The second claim follows, since for each [ n, n − r ] p -code there is an r × n parity check matrix, which has full rank. (cid:3) ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 13
Lemma 4.3.
Let d, r ∈ N and p prime. Then s A, ≤ d ( C rp ) − is equal to the maximum n such that there exists an [ n, n − r ] p -code of minimal distance at least d + 1 .Proof. By definition of s A, ≤ d ( C rp ) there exists a sequence S over C rp of length s A, ≤ d ( C rp ) − A -weighted zero-subsum of length at most d . Also note that the elements in S generate C rp ; this follows for example fromLemma 3.8. Thus, C S is an [ n, n − r ] p -code, and it cannot have a non-zero code-word of weight at most d ; that is, its minimal distance is at least d + 1. Conversely,given an [ n, n − r, d ′ ] p -code with d ′ > d , we get a sequence S C over C rp of length n whose shortest A -weighted zero-subsum has length d ′ > d . Thus, s A, ≤ d ( C rp ) > n ,completing the argument. (cid:3) We end this general section on the link between linear codes and A -weightedzero-sum problems by pointing out that the existence of m disjoint A -weightedzero-subsums of some sequence S corresponds precisely to the existence of m non-zero codewords in C S with pairwise disjoint supports. This fits together with anotion considered in coding theory, especially the case m = 2. We recall thatCohen and Lempel [8] called a linear code for which any two non-zero codewordsdo not have disjoint support an intersecting code. We refer to [28] for a moredetailed discussion of (binary) intersecting codes in the current context.5. Some exact values and bounds for s A, ≤ d ( C rp )We continue to use the notation that G denotes a finite abelian group and A = { , . . . , exp( G ) − } . In this section we establish various results for s A, ≤ d ( C rp ). Theproblem of determining this constant is equivalent to a problem in coding theory,as we saw in Section 4. Our results are mainly based on this link and results incoding theory. For the case that d = 3, it will however be advantageous to use afurther equivalence to reduce the problem to one in discrete geometry.We start by discussing some simple extremal cases. It is clear that for | G | 6 = 1we always have s A, ≤ ( G ) = ∞ . Moreover, by (4.1) we have that s A, ≤ d ( C rp ) = r + 1for d ≥ r + 1.In the following lemma we determine s A, ≤ ( C rp ) by a simple direct argument. Lemma 5.1.
Let p be a prime and let r ∈ N . Then s A, ≤ ( C rp ) = 1 + p r − p − .Proof. Let S be a sequence over C rp . Clearly, S has an A -weighted zero-subsumof length 1 if and only if S contains 0. So, suppose S contains only non-zeroelements. Moreover, we observe that S has an A -weighted zero-subsum if andonly if S contains two elements (including multiplicity) from the same (non-trivial)cyclic subgroup. Thus, the maximal length of a sequence without A -weighted zero-subsum of length at most 2 is equal to the number of non-trivial cyclic subgroupsof C rp , which equals p r − p − . This implies the claim. (cid:3) One could also use the link to coding theory to obtain this result—observe thatthe extremal examples correspond to the parity check matrix of p -ary Hammingcodes—yet in view of the simplicity of a direct argument we preferred to give it.We proceed to discuss the case that d = 3. It is well-known and not hard to seethat s A, ≤ ( C r ) = 1 + 2 r − for each r ≥ p >
2. Webegin by a further equivalent description of s A, ≤ ( C rp ). Recall that a cap set is a subset of an affine or projective space that does not contain three co-linear points.It is well-known that linear codes of minimal distance (at least) four and cap setsare related; see, e.g., [5, Section 4] or [25, Section 27.2]. We summarize this relationin a form convenient for our applications in the following lemma. We include thealready established relation of our problem on sequences to a problem on codes. Lemma 5.2.
Let p be an odd prime, and let r ≥ and n ≥ be integers. Let g , . . . , g n ∈ C rp \ { } and assume the g i ’s generate C rp . The following statementsare equivalent. (1) The sequence g . . . g n has no A -weighted zero-subsum of length at most . (2) The [ n, n − r ] p -code with parity check matrix [ g | · · · | g n ] has minimaldistance at least . (3) The set of points represented by the g i ’s in the projective space of dimension r − over the field with p elements is a cap set of size n .In particular, the following integers are equal. • s A, ≤ ( C rp ) − . • The maximal n such that there exists an [ n, n − r ] p -code of minimal distanceat least four. • The maximal cardinality of a cap set in the projective space of dimension r − over F p .Proof. The equivalence of the first two is merely Lemma 4.3. For the equivalencewith the third it suffices to note that a relation of the form ag i + bg j + cg k = 0with non-zero (modulo p ) coefficients a, b, c means that the points represented by g i , g j , g k are co-linear; in addition note that a relation of the form ag i + bg j = 0with non-zero (modulo p ) coefficients a, b would mean that g i and g j represent thesame point in the projective space, which is excluded by insisting that the size ofthe cap set is n . (cid:3) Having this equivalence at hand, there is wealth of results on s A, ≤ ( C rp ) available.We refer to [5] for a recent survey article on the problem of determining large capsets in projective spaces. However, only for small dimensions an answer for all p isknown. We summarize these results in the current notation. Theorem 5.3.
Let p be an odd prime. Then (1) s A, ≤ ( C p ) = 2 . (2) s A, ≤ ( C p ) = 3 . (3) s A, ≤ ( C p ) = 2 + p . (4) s A, ≤ ( C p ) = 2 + p . For larger values of r exact values are only known for small p ; some of these (for p ≤
7) are implicitly recalled in the subsequent results. We refer again to [5] formore complete information. Moreover, we point out that data on this problem canbe retrieved from databases for the parameters of codes; we mention specifically by Grassl [21] and MinT (see http://mint.sbg.ac.at ) bySch¨urer and Schmid [30].Indeed, recall from Lemma 4.3 that to determine s A, ≤ d ( C rp ) − n such that an [ n, n − r ] p -code with minimal distance greaterthan d exists. Or put differently, determining s A, ≤ d ( C rp ) is equivalent to determiningthe smallest n such that d is the largest minimum distance of an [ n, n − r ] p -code. ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 15
This type of information can be obtained from data on parameters of codes,though the particular piece of information we need is rather not highlighed in suchdata.Yet, note that, for some fixed p , having a table, such as given in ,with on one axis the length n of a code, on the other axis the dimension k of acode, and with entries the maximal minimal distance of an [ n, k ] p -code (or boundsfor it), to find information on s A, ≤ d for C rp we merely have to look along a suitablediagonal of the table, namely the one with n − k = r .Moreover, note that using the advanced user interface to MinT one can in factspecify as one of the search-parameters the co-dimension of the code, that is n − k = r (in our notation).The following results are extracted from the above mentioned data (except forthe results for large d that are merely (4.1)). We do not include results for groupsof rank one and two as these were already discussed in Lemma 5.1. However, wedo include the results for d = 2 and d = 3, already mentioned, to compare the sizeof the constants. Theorem 5.4. (1) s A, ≤ ( C ) = 14 , s A, ≤ ( C ) = 5 , and s A, ≤ d ( C ) = 4 for d ≥ . (2) s A, ≤ ( C ) = 41 , s A, ≤ ( C ) = 11 , s A, ≤ ( C ) = 6 , and s A, ≤ d ( C ) = 5 for d ≥ . (3) s A, ≤ ( C ) = 122 , s A, ≤ ( C ) = 21 , s A, ≤ ( C ) = 12 , s A, ≤ ( C ) = 7 , and s A, ≤ d ( C ) = 6 for d ≥ . (4) s A, ≤ ( C ) = 365 , s A, ≤ ( C ) = 57 , s A, ≤ ( C ) = 15 , s A, ≤ ( C ) = 13 , s A, ≤ d ( C ) = 8 ; and s A, ≤ d ( C ) = 8 for d ≥ . Theorem 5.5. (1) s A, ≤ ( C ) = 32 , s A, ≤ ( C ) = 7 , and s A, ≤ d ( C ) = 4 for d ≥ . (2) s A, ≤ ( C ) = 157 , s A, ≤ ( C ) = 27 , s A, ≤ ( C ) = 7 , and s A, ≤ d ( C ) = 5 for d ≥ . (3) s A, ≤ ( C ) = 782 , ≤ s A, ≤ ( C ) ≤ , s A, ≤ ( C ) = 13 , s A, ≤ ( C ) = 7 ,and s A, ≤ d ( C ) = 6 for d ≥ . Theorem 5.6. (1) s A, ≤ ( C ) = 58 , s A, ≤ ( C ) = 9 , and s A, ≤ d ( C ) = 4 for d ≥ . (2) s A, ≤ ( C ) = 400 , s A, ≤ ( C ) = 51 , s A, ≤ ( C ) = 9 , and s A, ≤ d ( C ) = 5 for d ≥ . In particular, for lager values of d also rather precise information for groupsof larger rank could be obtained in this way. However, we do not include thisinformation explicitly.We also include some information that we can obtain for p = 2; here, the problemreduces to the classical case. The general point is already established in [9]; wemerely added the numerical values using known data. We recall that in this casenot only is it known that s ≤ ( C r ) = 2 r but also that s ≤ ( C r ) = 1 + 2 r − (seeabove). Thus, we can reduce to considering d ≥ Theorem 5.7. (1) s ≤ ( C ) = 6 , s ≤ d ( C ) = 5 for d ≥ . (2) s ≤ ( C ) = 7 , s ≤ ( C ) = 7 , and s ≤ d ( C ) = 6 for d ≥ . (3) s ≤ ( C ) = 9 , s ≤ ( C ) = 8 , s ≤ ( C ) = 8 , and s ≤ ( C ) = 7 for d ≥ . (4) s ≤ ( C ) = 12 , s ≤ ( C ) = 10 , s ≤ ( C ) = 9 , s ≤ ( C ) = 9 , and s ≤ d ( C ) = 8 for d ≥ . (5) s ≤ ( C ) = 18 , s ≤ ( C ) = 13 , s ≤ ( C ) = 10 , s ≤ ( C ) = 10 , s ≤ ( C ) = 10 ,and s ≤ d ( C ) = 9 for d ≥ . (6) s ≤ ( C ) = 24 , s ≤ ( C ) = 19 , s ≤ ( C ) = 12 , s ≤ ( C ) = 11 , s ≤ ( C ) = 11 , s ≤ ( C ) = 11 , and s ≤ d ( C ) = 10 for d ≥ . (7) s ≤ ( C ) = 34 , s ≤ ( C ) = 25 , s ≤ ( C ) = 16 , s ≤ ( C ) = 13 , s ≤ ( C ) =12 , s ≤ ( C ) = 12 , s ≤ ( C ) = 12 , and s ≤ d ( C ) = 11 for d ≥ . We end the section by some additional discussion of the case p = 3 and d = 3,which is particularly popular. In this case only, we also include a somewhat detaileddiscussion of asymptotic bounds. We omit such a discussion in the general case;again we refer to [5] for additional information.As mentioned in the introduction Godinho, Lemos, and Marques [19] studiedsome plus-minus weighted zero-sum constants. For groups of exponent 3, the setsof weights { +1 , − } and { , . . . , exp( G ) − } are equivalent. Thus, our constant s A, ≤ ( C r ) coincides with their η A ( C r ); and as shown there (see Propositions 1 and2 in [19]) their other constants s A ( C r ) and g A ( C r ) can be expressed in terms of η A ( C r ), namely s A ( C r ) = g A ( C r ) = 2 η A ( C r ) − η A ( C ) and η A ( C ) in addition. Moreover, in the same way we can obtainthe bounds 113 ≤ η A ( C ) ≤ ≤ η A ( C ) ≤ ≤ η A ( C ) ≤ ≤ η A ( C ) ≤ r = 17, in part stemming from results on cap sets (see the respective entries inMinT for precise references).Moreover, using the link established in Lemma 5.2 and using lower bounds onthe size of caps in ternary spaces, we get that for sufficiently large r one has s A, ≤ ( C r ) ≥ . r and indeed one could take a slightly larger constant (see [5] for details). For large r this is considerably better than the bound given in [19] (see Propositions 3 and 4there). For an upper bound we recall that Bateman and Katz [3] recently showedthat the maximal size of a cap set in a ternary affine space of dimension r is O (3 r /r ǫ ) for some universal ǫ >
0. This is clearly also an upper bound for thesize of a cap set in a ternary projective space of dimension r − s A, ≤ ( C r ) = O (3 r /r ǫ ) . The gap between upper and lower bound is significant and even conjecturally itis not at all clear what should be the actual order of magnitude of s A, ≤ ( C r ) as r tends to infinity, while the problem, in the equivalent formulation for cap sets,received considerable attention.6. All multi-wise fully-weighted Davenport constants for somegroups
In the current section we establish the value of all fully-weighted multi-wiseconstants for certain groups, that is for some G and A = { , . . . , exp( G ) − } wedetermine D A,m ( G ) for each m . We make use of the results for s A, ≤ d ( G ) establishedbefore. The group we consider are on the one hand elementary p -groups of rank at ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 17 most two, and on the other hand C . We recall that a solution to this problem for C r for r ≤ Theorem 6.1.
Let p be a prime number. (1) D A,m ( C p ) = 2 m , in particular m A ( C p ) = 1 and D A, ( C p ) = 0 . (2) D A,m ( C p ) = 3 m for m ≤ ⌈ p/ ⌉ and D A,m ( C p ) = 2 m + ⌈ p/ ⌉ for m > ⌈ p/ ⌉ ,in particular m A ( C p ) = ⌈ p/ ⌉ and D A, ( C p ) = ⌈ p/ ⌉ .Proof. The statement for cyclic groups is a direct consequence of Lemmas 3.5, 3.6,and 5.1.For C p we note that s A, ≤ ( C p ) = p + 2 by Lemma 5.1 and s A, ≤ ( C p ) = 3. Thus, D A,m ( C p ) ≤ m for each m by Lemma 3.6 and D A,m ( C p ) ≥ m for 3 m ≤ p + 2 byLemma 3.5. Note that 3 m ≤ p + 2 is equivalent to m ≤ ⌈ p/ ⌉ .Now, suppose m > ⌈ p/ ⌉ . Write m = ℓ + ⌈ p/ ⌉ . We need to show D A,m ( C p ) =2 m + ⌈ p/ ⌉ , that is D A,m ( C p ) = 2 ℓ + 3 ⌈ p/ ⌉ . That this is a lower bound followsfrom D A, ⌈ p/ ⌉ ( C p ) = 3 ⌈ p/ ⌉ and Lemma 3.12. To show this is an upper bound,let S be a sequence of length 2 ℓ + 3 ⌈ p/ ⌉ . Since s A, ≤ ( C p ) = p + 2, the sequence S has (at least) ℓ disjoint A -weighted zero-subsums of length at most two; letus denote the corresponding sequences by T , . . . , T ℓ . Then the sequence R =( T . . . T ℓ ) − S has length at least 3 ⌈ p/ ⌉ . Thus, since s A, ≤ ( C p ) = 3, the sequence R has at least ⌈ p/ ⌉ disjoint A -weighted zero-subsums. Consequently, we haveat least ℓ + ⌈ p/ ⌉ = m disjoint A -weighted zero-subsums of S . This shows that D A,m ( C p ) ≤ m + ⌈ p/ ⌉ . (cid:3) To complement the result for elementary p -groups of rank at most two, we con-sider the problem for C . For the values of all the classical multi-wise Davenportconstants for C we refer to [4]. The result below shows an interesting phenome-non, namely that the difference between D A,m +1 ( G ) and D A,m ( G ) is not necessarilynon-increasing. Theorem 6.2.
We have D A, ( C ) = 4 , D A, ( C ) = 7 , D A, ( C ) = 9 , D A, ( C ) =12 , and D A,m ( C ) = 4+2 m for m ≥ . In particular, D A, ( C ) = 4 and m A ( C ) =5 .Proof. First, we use results we obtained earlier, to reduce the problem to showing D A, ( C ) ≥ D A, ( C ) ≤ D A, ( C ) = 4. By Lemma 3.5, with d = 3 and since s A, ≤ ( C ) =14, see Lemma 5.1, we get the lower bounds for D A,m ( C ) for m ∈ { , } . Then,by Lemma 3.12 we also get the lower bounds for each m ≥ s A, ≤ ( C ) = 5, it follows by Lemma 3.6 that D A, ( C ) ≤
7. Moreover, s A, ≤ ( C ) = 5 and Lemma 3.6 shows that D A, ( C ) = 9implies that D A, ( C ) ≤
12, and furthermore s A, ≤ ( C ) = 14, see Lemma 5.1, andLemma 3.6 then show D A,m ( C ) ≤ m for m ≥ D A, ( C ) ≥ D A, ( C ) ≤ D A, ( C ) ≥ e e e ( e + e )( e + e )( e + e ). It cannot have an A -weighted zero-subsum of length at most 2, thus if ithad two disjoint A -weighted zero-subsums they would both be of length 3. Oneof the two subsums has to contain at least two of e , e , e , say it contains e and e . Then the third element is necessarily e + e . However, the three otherelements e , ( e + e ) , ( e + e ) do not have a A -weighted zero-subsum. Thus, e e e ( e + e )( e + e )( e + e ) does not have two disjoint A -weighted zero-subsums.Consequently D A, ( C ) > D A, ( C ) ≤
9. We reduce this problem to the problem of checkingwhether three specific sequences have three disjoint A -weighted zero-subsums.First, we recall that any sequence of length 9 that contains 0 or an element morethan once or an element and its inverse, has an A -weighted subsum of length at mosttwo, and thus, since D A, ( C ) ≤
7, it has three disjoint A -weighted zero-subsums.Thus, we can restrict to considering squarefree sequences of length 9 where eachcyclic subgroup of C contains at most one element.Second, we recall that replacing an element occurring in a sequence by its inversehas no effect on the number of disjoint A -weighted zero-subsums.Thus, we can restrict to considering subsequences of e e e ( e + e )( e + e )( e + e ) ( e − e )( e − e )( e − e )( e + e + e )( e + e − e )( e − e + e )( e − e − e );this sequence has length 13 and contains one non-zero element from each cyclicsubgroup.A subsequence of length 9 is characterized by the 4 elements of the above 13that it does not contain. But, we do not need to check all sequences resulting fromomitting each possible choice of four elements, since the problem is invariant underisomorphisms of the group.We argue there are only three cases to consider. Let g , g , g , g be four distinctnon-zero elements (none the inverse of each other). While below we give a purelyalgebraic treatment, we remark that we could also consider this as a problem inthe two-dimensional ternary projective space; the three cases being four points ona line, three (yet not four) on a line, and no three co-linear points (i.e., a cap-set).Case 1: g , g , g , g do not generate C . In this case, we can assume that g = g + g and g = g − g . (Certainly, g , g can be written as a linearcombination of g , g and since we are free to choose signs, this is the only possibilitywe need to consider.)Case 2: g , g , g , g generate C , but there is some g j , say g , such that theset { g , g , g , g } \ { g } does not generate the group. We observe that g , g , g is a generating set; g , g are independent by assumption, while g is not in thesubgroup generated g and g . Moreover, g is an element of the group generatedby g and g , and as we can ignore signs, we can assume g = g − g .Case 3: g , g , g , g generate C , and in fact { g , g , g , g } \ { g j } generates thegroup for each j . We note that g , g , g is a generating set and g = a g + a g + a g . Since { g , g , g , g }\{ g j } is a generating set for each j it follows that all a i arenon-zero and therefore, as signs are irrelevant, we can assume g = − ( g + g + g ).Since isomorphisms preserve A -weighted zero-subsums we can choose for theindependent elements whatever independent elements we like. In case 1 we choose g = e and g = e . Thus after removing the four elements g , g , g , g thefollowing sequence remains: e ( e + e )( e + e )( e − e )( e − e )( e + e + e )( e + e − e )( e − e + e )( e − e − e ) . In case 2 and 3 we chose g = e − e + e and g = e + e − e and g = e − e − e . ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 19
Thus after removing the four elements g , g , g , g the following sequence re-mains in case 2: e e e ( e + e )( e + e )( e + e )( e − e )( e − e )( e + e + e ) . And, in case 3: e e e ( e + e )( e + e )( e − e )( e − e )( e − e )( e + e + e ) . Now, it remains to check that these three sequences of length 9 each have 3disjoint A -weighted zero-subsums. If this is established it follows from the abovearguments that in fact each sequence of length 9 over C has this property, andthus D A, ( C ) ≤ A -weighted subsums of length 3, where forclarity we write each element in parenthesis:( e ) + ( e + e ) + ( e − e ) , ( e + e ) + ( e + e + e ) + ( e − e + e ) , ( e − e ) + ( e + e − e ) + ( e − e − e )( e ) + ( e + e ) − ( e + e + e ) , ( e ) − ( e − e ) + ( e + e ) , ( e ) + ( e + e ) − ( e − e )( e ) + ( e ) − ( e + e ) , ( e ) + ( e + e ) − ( e − e ) , ( e − e ) + ( e − e ) − ( e + e + e ) (cid:3) Some asymptotic results
In the current section we complement the earlier results focused on small groupswith results focused on elementary p -groups of larger rank. Again, we use the linkto coding theory explained in Section 4.We use some ad-hoc terminology based on the one introduced in [28]. A function f : [0 , s ] → [0 , < s ≤
1, is called p -upper-bounding if it is non-increasing,continuous, and each [ n, k, d ] p -code with d/n in [0 , s ] satisfies kn ≤ f (cid:18) dn (cid:19) And, f is called asymptotically p -upper-bounding if kn . f (cid:0) dn (cid:1) for [ n, k, d ] p -codeswith sufficiently large n .Thus, (asymptotically) p -upper-bounding functions are the functions in the (as-ymptotic) upper bounds for the rate k/n of a p -linear code as a function of itsnormalized minimal distance d/n .The following is essentially a reformulation of Lemma 4.3, in a way more suitablefor the current applications. Lemma 7.1.
Let f be a p -upper-bounding function, and let d, n, r ∈ N with ≤ d ≤ n − and n − rn > f (cid:18) d + 1 n (cid:19) then s A, ≤ d ( C rp ) ≤ n . Moreover, the same holds true for f an asymptotically p -upper-bounding function and sufficiently large n such that the inequality holds uniformlyin n (that is n − rn > f (cid:0) d +1 n (cid:1) + ε for some ε > independent of n ). Proof.
Let S = g . . . g n be an arbitrary sequence over C rp . By Lemma 3.8 we knowthat s A, ≤ d ( C rp ) ≥ s A, ≤ d ( C sp ) for s ≤ r , and we thus can assume that the g i generate C rp .We choose some basis of C rp and C np . We apply Lemma 4.2 with the sequence S to get an [ n, n − r ] p -code C S ⊂ C np . Let e be the minimal distance of C S , i.e C S isan [ n, n − r, e ] p -code. But by assumption since f is upper-bounding and n − rn > f (cid:18) d + 1 n (cid:19) an [ n, n − r, d + 1] p -code cannot exist. This implies that e < d + 1, or equivalently d ≥ e . We conclude by applying Lemma 4.1, which shows that S possesses an A -weighted zero-sum subsequence of length e .The additional claim, for asymptotically upper-bounding functions, is immediatein view of the just given argument and the fact that our condition just negates n − rn . f (cid:0) d +1 n (cid:1) (cid:3) We recall the following well-known fact that we need in the subsequent arguments(see, e.g., [27, Appendix B.3]).
Lemma 7.2.
Let k, n ∈ N with n ≥ k . In an n -dimensional vector space over afield with p elements, the number of k -dimensional subspaces is equal to the p -arybinomial coefficient defined as (cid:20) nk (cid:21) p = ( p n − . . . ( p n − k +1 − p k − . . . ( p − Moreover, the number of k -dimensional subspaces containing a fixed j -dimensionalsubspace, k ≥ j , is equal to (cid:20) n − jk − j (cid:21) p . Now, we state one of the main results of this section, a lower bound for the fully-weighted multi-wise Davenport constants for elementary p -groups of large rank. Proposition 7.3.
Let m ∈ N and let p be a prime number. Then, for sufficientlylarge r , with A = { , . . . , p − } , D A,m ( C rp ) ≥ log p m log(1 + m ( p − r. Proof.
For m = 1 we know by (4.1) that D A,m ( C rp ) = r + 1. Since log p/ (log(1 +1( p − ≤
1, the claim follows. Now, we fix a positive integer m > m log p/ (log(1+ m ( p − > r is sufficiently large, there is an integer n such that r + m ≤ n < log p m log(1 + m ( p − r Recall that by Lemma 4.3 we can associate to each sequence S of length n over C rp whose elements generate C rp an [ n, n − r ] p -code, and indeed we can obtain everysuch code in this way.We observe that the condition that S has m disjoint A -weighted zero-sum sub-sequences translates to the condition that the associated code contains m non-zerocodewords c , . . . , c m such that pairwise intersections of their supports are empty.We call such a code m -inadmissible, otherwise it will be called m -admissible. ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 21
We first produce an upper bound on the total number of [ n, n − r ] p -codes thatare m -inadmissible.By definition any m -inadmissible code contains c , . . . , c m with the above men-tioned property. These c i ’s generate an m -dimensional vector space; to see this justnote that the non-zero coordinates of each c i are unique to that element and thusthe c m ’s are certainly independent.Let V denote the set of all subsets { d , . . . , d m } ⊂ C np \ { } such that the inter-section of the support of d u and d v is empty for all distinct u, v ∈ { , . . . , m } ; thus,in particular, all the d i ’s are distinct.We note that a code C is m -inadmissible if and only if V ⊂ C for some V ∈ V (this V is not necessarily unique). Moreover, Lemma 7.2 implies that for each V ∈ V there are (cid:2) n − mn − r − m (cid:3) p codes containing V ; to see this note that if V ⊂ C then C also contains the vector space generate by V , which is m -dimensional, and applyLemma 7.2. It thus follows that the total number of m -inadmissible codes cannotexceed |V| (cid:20) n − mn − r − m (cid:21) p . We give a simple estimate for |V| . An element { d , . . . , d m } of V can be described byspecifying for each l ∈ { , . . . , n } which of the supports of the d i ’s (if any) contains l and which (non-zero) value the respective coordinate has. Thus, for each l thereare 1 + m ( p −
1) possibilities, and consequently (1 + m ( p − n is an upper boundfor |V| .We therefore infer that the total number of m -inadmissible [ n, n − r ] p -codes isbounded above by (1 + m ( p − n (cid:20) n − mn − r − m (cid:21) p . Again, by Lemma 7.2, it follows that the ratio of total number of m -inadmissible[ n, n − r ] p -codes divided by total number of [ n, n − r ] p -codes is bounded above by(1 + m ( p − n (cid:2) n − mn − r − m (cid:3) p (cid:2) nn − r (cid:3) p = (1 + m ( p − n n Y k = n − m +1 p k − r − p k − ≤ (1 + m ( p − n n Y k = n − m +1 p k − r p k = (1 + m ( p − n p − rm = p n log p (1+ m ( p − − rm Thus, it follow that as soon as ( n log p (1 + m ( p − − rm ) is negative, that is nr < m log p (1 + m ( p − D A,m ( C rp ) ≥ log p m log(1 + m ( p − r for sufficiently large r . (cid:3) The following result combines Lemmas 3.6 and 7.1.
Lemma 7.4.
Let m, r ∈ N and let p be a prime number, and let A = { , · · · , p − } .Furthermore, let f be an asymptotic upper-bounding function. (1) If D A,m ( C rp ) ≤ br for each sufficiently large r and c is a solution to theinequality b + c − b + c > f (cid:18) cb + c (cid:19) , then for each sufficiently large integer r , we have D A,m +1 ( C rp ) ≤ ( b + c ) r . (2) If D A,m ( C rp ) . br and c is a solution to the inequality b + c − b + c ≥ f (cid:18) cb + c (cid:19) , then we have D A,m +1 ( C rp ) . ( b + c ) r .Proof. We start by proving 1. Given the assumptions, we have( b + c ) r − r ( b + c ) r = b + c − b + c > f (cid:18) cb + c (cid:19) = f (cid:18) cr ( b + c ) r (cid:19) ≥ f (cid:18) cr + 1( b + c ) r (cid:19) , where we used that f is non-increasing. Clearly, the inequality holds uniformly in r . Replacing ⌊ ( b + c ) r ⌋ by n and ⌊ cr ⌋ by d , we obtain (for r sufficiently large andby the continuity of f ) that n − rn > f (cid:18) d + 1 n (cid:19) . And, the inequality still holds uniformly. By Lemma 7.1 we have s A, ≤⌊ cr ⌋ ( C rp ) ≤ n = ⌊ ( b + c ) r ⌋ ≤ ( b + c ) r and then by Lemma 3.6 D A,m +1 ( C rp ) ≤ min i ∈ N max { D A,m ( C rp ) + i, s A, ≤ i ( C rp ) } . Finally, D A,m +1 ( C rp ) ≤ min i ∈ N max { D A,m ( C rp ) + i, s A, ≤ i ( C rp ) }≤ max { br + ⌊ cr ⌋ , s A, ≤⌊ cr ⌋ ( C rp ) }≤ max { ( b + c ) r, ( b + c ) r } = ( b + c ) r showing the claim in 1.Now, let ε > b + c − p + c ≥ f ( cb + c ), we get that (the left hand-side increases while the right-hand side does notincrease) b + c + ε/ − b + c + ε/ > b + c − b + c ≥ f (cid:18) cb + c (cid:19) ≥ f (cid:18) c + ε/ b + c + ε/ (cid:19) . As above we thus get, for sufficiently larger r , that s A, ≤⌊ ( c + ε/ r ⌋ ( C rp ) ≤ n = ⌊ ( b + ( c + ε/ r ⌋ ≤ ( b + ( c + ε/ r. A sequence S of length at least ( b + c + ε ) r thus contains a subsequence T oflength at most ( c + ε/ r that has 0 as an A -weighted sum. Since the length of T − S is at least ( b + ε/ r and since we assumed D A,m ( C rp ) . br it follows that(if r is sufficiently large) the sequence T − S admits m disjoint A -weighted zero-subsums. Thus, S admits m + 1 disjoint A -weighted zero-subsums, showing that ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 23 D A,m +1 ( C rp ) ≤ ( b + c + ε ) r for all sufficiently large r . Consequently, D A,m +1 ( C rp ) . ( b + c ) r . (cid:3) We use the just established lemma in two ways. First, we give bounds for D A,m ( C rp ) for small p and m yet large r . Then, in Theorem 7.6 we investigate theasymptotic behavior of D A,m ( C rp ) for large m and r ; recall that we studied theproblem for fixed r and large m in Theorem 3.2.Of course, to apply Lemma 7.4 we need some asymptotic p -upper boundingfunction. We recall some asymptotic bounds on the parameters of codes that weuse (see for example [26, Section 2.10]). For a prime p and 0 < x ≤ ( p − /p , let h p ( x ) = − x log p ( x/ ( p − − (1 − x ) log p (1 − x )and h p ( x ) = 0, denote the p -ary entropy function. The following functions are p -upper bounding functions on [0 , ( p − /p ]:(1) 1 − h p (cid:16) x (cid:17) by the asymptotic Hamming bound.(2) 1 − h p p − p − s p − p (cid:18) p − p − x (cid:19)! by the asymptotic Elias bound.(3) h p p − − ( p − x − p ( p − x (1 − x ) p ! by the first MRRW bound.We now formulate the result for small p and m ; as the proof shows, we couldobtain similar results for further values. We recall from (4.1) that D A, ( C rp ) = r +1,which is why we do not include this case. Moreover, the case p = 2 was consideredin [28] and we do not repeat the result. Theorem 7.5.
For each sufficiently large integer r we have: (1) 1 . r ≤ D A, ( C r ) ≤ . r . r ≤ D A, ( C r ) ≤ . r r ≤ D A, ( C r ) ≤ . r . r ≤ D A, ( C r ) ≤ . r (2) 1 . r ≤ D A, ( C r ) ≤ . r . r ≤ D A, ( C r ) ≤ . r . r ≤ D A, ( C r ) ≤ . r . r ≤ D A, ( C r ) ≤ . r (3) 1 . r ≤ D A, ( C r ) ≤ . r . r ≤ D A, ( C r ) ≤ . r . r ≤ D A, ( C r ) ≤ . r . r ≤ D A, ( C r ) ≤ . r Proof.
The lower bounds are merely derived from Proposition 7.3, rounding down the exact value. For the upper bounds we apply repeatedly Lemma 7.4. Since D A, ( C rp ) = r + 1, for any fixed b >
1, we have D A, ( C rp ) < br ; we take b = 1 .
001 asstarting value. Numerically, we find a solution c f to the inequality b + c − b + c > f ( cb + c )for f one of the upper-bounding functions mentioned above; in practice we find asolution of the inequality and round it up . We then know D A, ( C rp ) ≤ ( b + c ,f ) r .For f the upper-bounding function that yields the smallest c ,f , we set b = b + c ,f .(In fact, in this case this is always the first MRRW bound but later it can also bethe asymptotic Elias bound.) We have D A, ( C rp ) ≤ b r and this is the bound wegive in the result. We then proceed in the same way to get a bound for D A, ( C rp ),and so on. (cid:3) We continue by investigating the behavior of D A,m ( C rp ) for large m and r . Theorem 7.6.
Let p be a prime number and A = { , · · · , p − } . When m tendsto infinity, we have lim sup r → + ∞ D A,m ( C rp ) r . p m log m . Proof.
We apply Lemma 7.4 with the function1 − h p (cid:16) x (cid:17) , which is an asymptotic p -upper bounding function by the asymptotic Hammingbound (see above).Recursively, we define a sequence ( v m ) m ∈ N . We set v = 1 and we define v m +1 as the smallest positive real such that (where V m = v + · · · + v m )(7.1) 1 V m + v m +1 = h p (cid:18) v m +1 V m + v m +1 ) (cid:19) . We note that this is well-defined and that v m ≤ p − /p for each m ; recall that h p is convex and attains its maximum of 1 at ( p − /p .Note that by Lemma 7.4 D A,m ( C rp ) . V m r . Observe that from this and Propo-sition 7.3 it follows that V m ≫ m/ log m for m → ∞ .We proceed to investigate the thus defined quantities. Multiplying (7.1) by(2 log p )( V m + v m +1 ) /v m +1 , gives2 log pv m +1 = − log (cid:18) v m +1 V m + v m +1 )( p − (cid:19) + (cid:18) − V m + v m +1 ) v m +1 (cid:19) log (cid:18) − v m +1 V m + v m +1 ) (cid:19) = − log (cid:18) v m +1 V m + v m +1 )( p − (cid:19) + O (1) ULTI-WISE AND CONSTRAINED FULLY WEIGHTED DAVENPORT CONSTANTS 25 where we used that (1 − y )(log(1 − y − )) = O (1) for y → ∞ and that v m is boundedwhile V m → ∞ as m → ∞ . Consequently, for the second equality using again that v m is bounded while V m → ∞ as m → ∞ ,2 log pv m +1 = log( V m + v m +1 ) − log( v m +1 ) + O (1) = log( V m ) − log( v m +1 ) + O (1) . If follows that, as m tends to infinity,(7.2) v m +1 ∼ p log V m . Using again V m ≫ m/ log m , it follows that V m +1 − V m = v m +1 . p log m and therefore, summing all these estimates yields V m . p m − X k =1 k ∼ p m log m establishing the claimed upper bound. (cid:3) Combining the lower and the upper bound for D A,m ( C rp ) from Proposition 7.3and the theorem above we get thatlog p m log m . lim sup r → + ∞ D A,m ( C rp ) r . p m log m . The lower bound seems more likely to give the correct growth. For some discussionof this in the case of p = 2, we refer to [28]. Acknowledgment
The authors would like to thank the referees for numerous detailed remarks andsuggestions.
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