The Davenport constant of a box
Salvatore Tringali
(based on joint work with Alain Plagne: arXiv:1405.4363)
Séminaire Caesar – Sep 18, 2014
Centre de mathématiques Laurent Schwartz, X - Palaiseau, Sep 18, 2014
Introduction
Basic definitions
Unordered sequences
Given a set X , we let X ∗ be the “abelian Kleene star” of X and by F (X ) = (X ∗ , ·) the
free abelian monoid over X , which we write multiplicatively (we use ε for the identity)
We think of a non-empty word in X ∗ as an unordered sequence and we write it as
x1 · · · xn (if there is no ambiguity)
We have a (monoid) homomorphism k · k : F (X ) → (N, +) given by kεk := 0 and
ksk := n for s = x1 · · · xn ∈ X ∗ \ {ε}
For s ∈ X ∗ we refer to ksk as the length of s
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Introduction
Basic definitions
Zero-sum sequences with support in a set
Let G = (G , +) be an additively written abelian group and assume X ⊆ G
The function π : X ∗ → G taking ε to 0 and a non-empty sequence x1 · · · xn to
x1 + · · · + xn is then a (monoid) homomorphism F (X ) → G
We use B(X ) for the kernel of π, and we refer to the elements of B(X ) as the
zero-sum sequences (shortly, ZSSs) of G with support in X
A non-empty sequence x1 · · · xn ∈ X ∗ belongs to B(X ) iff
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Pn
i=1
xi = 0
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Introduction
Basic definitions
Minimal zero-sum sequences
A non-empty sequence s = x1 · · · xn ∈ B(X
P ) is called minimal if there does not exist
any non-empty set I ( {1, . . . , n} s.t.
i∈I xi = 0
The set of all minimal ZSSs of G with support in X will be denoted by A(X ): this is the
set of atoms of B(X ) regarded as a submonoid of F (X )
[If M = (M, ·) is a multiplicatively written monoid, an element a ∈ M is an atom of M if
a∈
/ M× and a = bc for some b, c ∈ M implies b ∈ M× or c ∈ M× ]
We define the Davenport constant of G relative to X , here denoted by D(X ), as the
sup of kak for a ∈ A(X )
In particular, we refer to D(G ) as the Davenport constant of G
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Introduction
Goals, motivations and achievements
Goals and motivations
Compute or estimate D(X ) in the case when G ∼
= H × (Z, +)d , with d ∈ N+ and
H = (H, +) a finite abelian group, and X is a box (or rectangle), viz. a set of the
form X0 × X1 × · · · × Xd , with X0 ⊆ H and Xi = J−mi , Mi K for some mi , Mi ∈ N
When H is trivial, this is motivated by the study of direct-sum decompositions in module
theory and fits into a research from outlined in the final section of:
N. R. Baeth and A. Geroldinger
Monoids of modules and arithmetic of direct-sum decompositions
Pacific J. Math., to appear (arXiv:1401.6553)
More in general, D(·) is a crucial invariant describing the arithmetic of certain monoids
arising from the study of non-unique factorization in the ring of integers of a number field
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Introduction
Goals, motivations and achievements
What we did...
We derive
(i) elementary (upper and lower) bounds on D(X ) when G = (Z, +) along with an
asymptotic bound in the special case when X = J−m, MK for some m, M ∈ N
(ii) an explicit description of the ZSSs of B(J−m, mK) of maximal (i.e., 2m − 1) or
almost maximal (i.e., 2m − 2) length when G = (Z, +) and m is an integer ≥ 2
(iii) “sharp” bounds on D(X ) when G = (Z, +)d and X is a symmetric box (i.e.,
X = J−m1 , m1 K × · · · × J−md , md K for some m1 , . . . , md ∈ N)
(iv) some multiplicative properties of D(·)
In addition to this, we have some questions
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Introduction
Goals, motivations and achievements
...and what we knew
Point (iii) improves on an upper bound obtained in:
P. Diaconis, R. Graham, and B. Sturmfels, “Primitive partition identities”
in: D. Miklós, V. T. Sós, and T. Szõnyi (eds.)
Combinatorics: Paul Erdős is eighty, Vol. 2
János Bolyai Math. Soc., 1996, 173–192
In addition, work by P. A. Sissokho and coauthors relative to the case G = (Z, +), e.g.
M. L. Sahs, P. A. Sissokho, and J. N. Torf
A zero-sum theorem over Z
Integers 13 (2013), #A71
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Theorems and proofs
Results in dimension 1
Bounds for the case G = (Z, +)
Theorem 1. We have χ(X ) ≤ D(X ) ≤ diam(X ), where
diam(X ) := sup |x − y |
and
χ(X ) =
x,y ∈X
sup
x,y ∈X :xy ≤0
|x| + |y |
gcd(x, y )
(with the convention that sup(∅) := 0 and 0/0 := gcd(0, 0) := 1)
The above inequality is sharp:
Corollary 1. If m and M are non-negative integers, then
m+M
≤ D(J−m, MK) ≤ m + M.
gcd(m, M)
In particular, if m and M are coprime, then D(J−m, MK) = m + M.
It follows that for an integer m ≥ 0 it holds

 1
2
D(J−m, mK) =

2m − 1
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if m = 0
if m = 1
ifm ≥ 2
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Theorems and proofs
Results in dimension 1
Asymptotic bounds for the case G = (Z, +)
We have the following asymptotic result:
Theorem 2. Let m, M ∈ N+ , m ≤ M. Then, for any real θ > 0 it holds that
D(J−m, MK) = m + M + o(mθ )
as m → ∞
In the preprint this is proved only for exponents θ ≥ 0.525, a value resulting from the
application of Hoheisel’s theorem on the existence of a prime in intervals of the form
(x − x θ , x), as improved in:
R. C. Baker, G. Harman, and J. Pintz
The difference between consecutive primes, II
Proc. London Math. Soc. 83 (2001), 532–562
Later, we realized that this can be extended to any exponent θ > 0 by the following:
Lemma 1. For every θ ∈ [0, 1[ there exists v ∈ N such that for all a, b ∈ N with
b ≥ a ≥ v there exist coprime integers h ∈ [a − aθ , a] and k ∈ [b − aθ , b]
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Theorems and proofs
Results in dimension 1
Inverse theorems for the case G = (Z, +)
We have the following inverse theorems:
Theorem 3. Let m and M be positive integers, and let s ∈ B(J−m, MK) be a ZSS of
length m + M. Then, s is minimal iff gcd(m, M) = 1 and s = M m · (−m)M
This leads to the following (well-known) result:
Corollary 2. Let m be an integer ≥ 2 and s ∈ B(J−m, mK) be a ZSS of length
2m − 1. Then, s is minimal iff s = mm−1 · (−m + 1)m or s = (−m)m−1 · (m − 1)m
The next theorem is somewhat more complicated:
Theorem 4. Let m be an integer ≥ 3, and let s ∈ B(J−m, mK) be a ZSS of length
2m − 2. Then, s is minimal iff one of the following holds:
(i) m is odd and either s = mm−2 · (−m + 2)m or s = (−m)m−2 · (m − 2)m
(ii) s = mm−2 · (−m + 1)m−1 · 1 or s = (−m)m−2 · (m − 1)m−1 · (−1)
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Theorems and proofs
Results in dimension 1
A couple of technical lemmas
Our proofs of Theorems 1, 3 and 4 are essentially based on the following lemmas
Lemma 2. Let x1 · · · xn ∈ A(G ), and assume there exist
P a set S and a permutation σ
of J1, nK such that for any i ∈ J1, nK, the partial sum il=1 xσ(l) belongs to S. Then,
n ≤ |S|. Furthermore, if n ≥ 3, xσ(2) 6= xσ(3) and xσ(2) + xσ(3) ∈ S, then n ≤ |S| − 1.
For S = G this implies the classical result according to which the Davenport constant of
a finite cyclic group of order n is n (...)
Lemma 3. Assume X is finite and non-empty, and let s = x1 · · · xn be a minimal ZSS
of B(X ) of length n ≥ 2. There then exists a permutation σ of J1, nK such that
P
xσ(i) i−1
l=1 xσ(l) < 0
for every i ∈ J1, n − 1K. In particular,
min(X ) ≤
i
X
xσ(l) ≤ max(X )
l=1
for every i ∈ J1, nK, where the inequality on the left (resp., on the right) holds as an
equality only if xσ(1) = min(X ) (resp., xσ(1) = max(X )).
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Theorems and proofs
Results in dimension ≥ 2
Bounds for the case G = (Z, +)d
The following results give bounds on D(X ) when X is a symmetric box of Zd (d ∈ N+ )
Theorem 5. Let m1 , . . . , md be non-negative integers, and set k := d +
D(J−m1 , m1 K × · · · × J−md , md K) ≤
d
Y
1
d
− 1. Then,
(2mi k + 1)
(1)
i=1
More in particular, if m is a non-negative integer, then
(2m − 1 + 2δm,0 + δm,1 )d ≤ D(J−m, mKd ) ≤ (2mk + 1)d
(2)
where δi,j is a Kronecker delta
For d = 2 the 2nd inequality in Theorem 5 can be improved to
D(J−m, mKd ) ≤ (2m + 1)(4m + 1)
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Theorems and proofs
Results in dimension ≥ 2
Some words on the proof
(1) and (3) are essentially based on a connection (first noticed by Diaconis, Graham and
Sturmfels) between the Davenport constant and (a generalization of) the Steinitz
constant of a finite-dimensional real normed space, see in particular
2
W. Banaszczyk, The Steinitz lemma in l∞
Period. Math. Hungar. 22 (1991), 183–186
On the other hand, the inequality on the left of Equ. (2) is based on the explicit
construction of a ZSS sd of length (2m − 1 + 2δm,0 + δm,1 )d
In fact, sd is constructed recursively as follows (we focus on the case m ≥ 2): If we write
sd = x1 · · · xn , where n = (2m − 1)d and xi ∈ J−m, mKd , then
sd+1 := (x1 , m)m−1 · · · (xn , m)m−1 · (0, −m + 1)mn
As a base of induction we consider d = 2, for which we have s2 = mm−1 (−m + 1)m
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Theorems and proofs
Miscellaneous results
Multiplicative properties of D(·)
In certain cases, D(·) behaves as a submultiplicative function:
Theorem 6. Let G = G1 × G2 , where Gi = (Gi , +) is an abelian group, and let
X2 ⊆ G2 . Then,
D(G1 × X2 ) ≤ D(G1 ) D(X2 )
On the other hand, there’re cases when D is supermultiplicative:
Theorem 7. Let G = H × (Z, +)d , where H = (H, +) is a finite abelian group and
d ∈ N+ , and let X = H × J−m, mKd , where m is a non-negative integer. Then,
D(X ) ≥ D(H) (2m − 1 + 2δm,0 + δm,1 )d
The above theorems together imply the following:
Corollary 3. Let G = H × (Z, +), where H is a finite cyclic group and m ∈ N. Then,
D(H × J−m, mK) = D(H) D(J−m, mK)
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Question time
“Il faut ecouter plûtot que parler”
Some questions
Here are some questions for which we do not know an answer:
(i) Let G = (Z, +). Can you find at least one case when D(X ) 6= χ(X )?
(ii) Let G = (Z, +)d and m1 , . . . , md ∈ N. Can you prove or disprove that
D(J−m1 , m1 K × · · · × J−md , md K) =
d
Y
i=1
D(J−mi , mi K)?
(iii) Is there a “nice characterization” of the cases in which Theorem 6 can be extended
to prove that D(X1 × X2 ) ≤ D(X1 ) D(X2 ) for all finite sets X1 ⊆ G1 and X2 ⊆ G2 ?
(iv) Can Corollary 7 be extended to prove that if G = H × (Z, +)d , d ∈ N+ , and
X = H × B for a symmetric box B ⊆ Zd then D(H × B) = D(H) × D(B)?
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Thanks
On ne voit bien qu’avec le coeur
Merci bien pour votre attention !
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