Bounding some constants in quantitative
non-unique factorization theory via coding
theory
W.A. Schmid1
Group MTII at LAGA a joint lab of the CNRS and Universities Paris 13 and 8
Jan 2013 / San Diego
1
Supported by the ANR CAESAR
A question of quantitative non-unique factorization
theory
(generalized) Prime Number Theorem:
There are approx.
x
(log log x)j−1
log x
natural numbers below x that are the product of (at most) j
primes.
A generalization (Halter-Koch):
Let OK be the ring of integers of some number field K , with
class group G. In OK there are approx.
x
(log log x)Dj (G)−1
log x
non-associated irreducible elements of absolute norm at most
x that are the product of at most j irreducible elements [in the
sense, cannot be written as the product of j + 1 or more irred.
elements, for a certain integer Dj (G) that can be described in
terms of the class group.
A question of quantitative non-unique factorization
theory
(generalized) Prime Number Theorem:
There are approx.
x
(log log x)j−1
log x
natural numbers below x that are the product of (at most) j
primes.
A generalization (Halter-Koch):
Let OK be the ring of integers of some number field K , with
class group G. In OK there are approx.
x
(log log x)Dj (G)−1
log x
non-associated irreducible elements of absolute norm at most
x that are the product of at most j irreducible elements [in the
sense, cannot be written as the product of j + 1 or more irred.
elements, for a certain integer Dj (G) that can be described in
terms of the class group.
Zero-sum problems / Davenport constant
Let (G, +, 0) be a finite abelian group.
Let S = g1 . . . gn be a sequence of elements of G.
Simple fact: If n is large enough, there exists a non-empty
I ⊂ [1, n] such that
X
gi = 0.
i∈I
‘If S is sufficiently long, then it has a zero-sum subsequence.’
Zero-sum problems / Davenport constant
Let (G, +, 0) be a finite abelian group.
Let S = g1 . . . gn be a sequence of elements of G.
Simple fact: If n is large enough, there exists a non-empty
I ⊂ [1, n] such that
X
gi = 0.
i∈I
‘If S is sufficiently long, then it has a zero-sum subsequence.’
Zero-sum problems / Davenport constant
Let (G, +, 0) be a finite abelian group.
Let S = g1 . . . gn be a sequence of elements of G.
Simple fact: If n is large enough, there exists a non-empty
I ⊂ [1, n] such that
X
gi = 0.
i∈I
‘If S is sufficiently long, then it has a zero-sum subsequence.’
Davenport constant
For (G, +) finite abelian group. Let D(G) denote the Davenport
constant, i.e.,
I
the smallest ` such that each sequence g1 . . . g` over G
has
P a (non-empty) zero-sum subsequence, i.e.,
6 I ⊂ {1, . . . `}.
i∈I gi = 0 for some ∅ =
I
equivalently, 1 plus the maximal length of a zero-sum free
sequence.
I
equivalently, thePmaximal lengthP
of a minimal zero-sum
`
sequence, i.e., i=1 gi = 0 yet i∈I gi 6= 0 for
∅=
6 I ( {1, . . . `}.
Studied since the mid 1960s.
Motivated by questions of Baayen, Davenport, Erdős.
Various applications: Number Theory (Carmichael numbers,
Factorizations), Graph Theory
Davenport constant
For (G, +) finite abelian group. Let D(G) denote the Davenport
constant, i.e.,
I
the smallest ` such that each sequence g1 . . . g` over G
has
P a (non-empty) zero-sum subsequence, i.e.,
6 I ⊂ {1, . . . `}.
i∈I gi = 0 for some ∅ =
I
equivalently, 1 plus the maximal length of a zero-sum free
sequence.
I
equivalently, thePmaximal lengthP
of a minimal zero-sum
`
sequence, i.e., i=1 gi = 0 yet i∈I gi 6= 0 for
∅=
6 I ( {1, . . . `}.
Studied since the mid 1960s.
Motivated by questions of Baayen, Davenport, Erdős.
Various applications: Number Theory (Carmichael numbers,
Factorizations), Graph Theory
Davenport constant
For (G, +) finite abelian group. Let D(G) denote the Davenport
constant, i.e.,
I
the smallest ` such that each sequence g1 . . . g` over G
has
P a (non-empty) zero-sum subsequence, i.e.,
6 I ⊂ {1, . . . `}.
i∈I gi = 0 for some ∅ =
I
equivalently, 1 plus the maximal length of a zero-sum free
sequence.
I
equivalently, thePmaximal lengthP
of a minimal zero-sum
`
sequence, i.e., i=1 gi = 0 yet i∈I gi 6= 0 for
∅=
6 I ( {1, . . . `}.
Studied since the mid 1960s.
Motivated by questions of Baayen, Davenport, Erdős.
Various applications: Number Theory (Carmichael numbers,
Factorizations), Graph Theory
Davenport constant
For (G, +) finite abelian group. Let D(G) denote the Davenport
constant, i.e.,
I
the smallest ` such that each sequence g1 . . . g` over G
has
P a (non-empty) zero-sum subsequence, i.e.,
6 I ⊂ {1, . . . `}.
i∈I gi = 0 for some ∅ =
I
equivalently, 1 plus the maximal length of a zero-sum free
sequence.
I
equivalently, thePmaximal lengthP
of a minimal zero-sum
`
sequence, i.e., i=1 gi = 0 yet i∈I gi 6= 0 for
∅=
6 I ( {1, . . . `}.
Studied since the mid 1960s.
Motivated by questions of Baayen, Davenport, Erdős.
Various applications: Number Theory (Carmichael numbers,
Factorizations), Graph Theory
Davenport constant
For (G, +) finite abelian group. Let D(G) denote the Davenport
constant, i.e.,
I
the smallest ` such that each sequence g1 . . . g` over G
has
P a (non-empty) zero-sum subsequence, i.e.,
6 I ⊂ {1, . . . `}.
i∈I gi = 0 for some ∅ =
I
equivalently, 1 plus the maximal length of a zero-sum free
sequence.
I
equivalently, thePmaximal lengthP
of a minimal zero-sum
`
sequence, i.e., i=1 gi = 0 yet i∈I gi 6= 0 for
∅=
6 I ( {1, . . . `}.
Studied since the mid 1960s.
Motivated by questions of Baayen, Davenport, Erdős.
Various applications: Number Theory (Carmichael numbers,
Factorizations), Graph Theory
Some results on D(G)
Let G = Cn1 ⊕ · · · ⊕ Cnr with ni | ni+1 . Then,
D(G) ≥ 1 +
r
X
(ni − 1) = D∗ (G).
i=1
Equality holds for (Olson, Kruyswijk, van Emde Boas, 1969)
I p-groups (group rings, later polynomial method).
I groups of rank at most 2 (inductive method, reduction to
p-groups).
In some other cases, e.g.,
I C 2 ⊕ C2n (van Emde Boas).
2
I C 2 ⊕ C3n (Bhowmik, Schlage-Puchta).
3
I C 2 ⊕ C4n and C 2 ⊕ C6n (S.).
4
6
But, not always. For example (Baayen), for odd n,
D(C24 ⊕ C2n ) > D∗ (C24 ⊕ C2n ).
Numerous other examples by Geroldinger et al. (but none for
groups of rank three or Cnr ).
Some results on D(G)
Let G = Cn1 ⊕ · · · ⊕ Cnr with ni | ni+1 . Then,
D(G) ≥ 1 +
r
X
(ni − 1) = D∗ (G).
i=1
Equality holds for (Olson, Kruyswijk, van Emde Boas, 1969)
I p-groups (group rings, later polynomial method).
I groups of rank at most 2 (inductive method, reduction to
p-groups).
In some other cases, e.g.,
I C 2 ⊕ C2n (van Emde Boas).
2
I C 2 ⊕ C3n (Bhowmik, Schlage-Puchta).
3
I C 2 ⊕ C4n and C 2 ⊕ C6n (S.).
4
6
But, not always. For example (Baayen), for odd n,
D(C24 ⊕ C2n ) > D∗ (C24 ⊕ C2n ).
Numerous other examples by Geroldinger et al. (but none for
groups of rank three or Cnr ).
Some results on D(G)
Let G = Cn1 ⊕ · · · ⊕ Cnr with ni | ni+1 . Then,
D(G) ≥ 1 +
r
X
(ni − 1) = D∗ (G).
i=1
Equality holds for (Olson, Kruyswijk, van Emde Boas, 1969)
I p-groups (group rings, later polynomial method).
I groups of rank at most 2 (inductive method, reduction to
p-groups).
In some other cases, e.g.,
I C 2 ⊕ C2n (van Emde Boas).
2
I C 2 ⊕ C3n (Bhowmik, Schlage-Puchta).
3
I C 2 ⊕ C4n and C 2 ⊕ C6n (S.).
4
6
But, not always. For example (Baayen), for odd n,
D(C24 ⊕ C2n ) > D∗ (C24 ⊕ C2n ).
Numerous other examples by Geroldinger et al. (but none for
groups of rank three or Cnr ).
Some results on D(G)
Let G = Cn1 ⊕ · · · ⊕ Cnr with ni | ni+1 . Then,
D(G) ≥ 1 +
r
X
(ni − 1) = D∗ (G).
i=1
Equality holds for (Olson, Kruyswijk, van Emde Boas, 1969)
I p-groups (group rings, later polynomial method).
I groups of rank at most 2 (inductive method, reduction to
p-groups).
In some other cases, e.g.,
I C 2 ⊕ C2n (van Emde Boas).
2
I C 2 ⊕ C3n (Bhowmik, Schlage-Puchta).
3
I C 2 ⊕ C4n and C 2 ⊕ C6n (S.).
4
6
But, not always. For example (Baayen), for odd n,
D(C24 ⊕ C2n ) > D∗ (C24 ⊕ C2n ).
Numerous other examples by Geroldinger et al. (but none for
groups of rank three or Cnr ).
Some results on D(G)
Let G = Cn1 ⊕ · · · ⊕ Cnr with ni | ni+1 . Then,
D(G) ≥ 1 +
r
X
(ni − 1) = D∗ (G).
i=1
Equality holds for (Olson, Kruyswijk, van Emde Boas, 1969)
I p-groups (group rings, later polynomial method).
I groups of rank at most 2 (inductive method, reduction to
p-groups).
In some other cases, e.g.,
I C 2 ⊕ C2n (van Emde Boas).
2
I C 2 ⊕ C3n (Bhowmik, Schlage-Puchta).
3
I C 2 ⊕ C4n and C 2 ⊕ C6n (S.).
4
6
But, not always. For example (Baayen), for odd n,
D(C24 ⊕ C2n ) > D∗ (C24 ⊕ C2n ).
Numerous other examples by Geroldinger et al. (but none for
groups of rank three or Cnr ).
Some results on D(G)
Let G = Cn1 ⊕ · · · ⊕ Cnr with ni | ni+1 . Then,
D(G) ≥ 1 +
r
X
(ni − 1) = D∗ (G).
i=1
Equality holds for (Olson, Kruyswijk, van Emde Boas, 1969)
I p-groups (group rings, later polynomial method).
I groups of rank at most 2 (inductive method, reduction to
p-groups).
In some other cases, e.g.,
I C 2 ⊕ C2n (van Emde Boas).
2
I C 2 ⊕ C3n (Bhowmik, Schlage-Puchta).
3
I C 2 ⊕ C4n and C 2 ⊕ C6n (S.).
4
6
But, not always. For example (Baayen), for odd n,
D(C24 ⊕ C2n ) > D∗ (C24 ⊕ C2n ).
Numerous other examples by Geroldinger et al. (but none for
groups of rank three or Cnr ).
Some results on D(G)
Let G = Cn1 ⊕ · · · ⊕ Cnr with ni | ni+1 . Then,
D(G) ≥ 1 +
r
X
(ni − 1) = D∗ (G).
i=1
Equality holds for (Olson, Kruyswijk, van Emde Boas, 1969)
I p-groups (group rings, later polynomial method).
I groups of rank at most 2 (inductive method, reduction to
p-groups).
In some other cases, e.g.,
I C 2 ⊕ C2n (van Emde Boas).
2
I C 2 ⊕ C3n (Bhowmik, Schlage-Puchta).
3
I C 2 ⊕ C4n and C 2 ⊕ C6n (S.).
4
6
But, not always. For example (Baayen), for odd n,
D(C24 ⊕ C2n ) > D∗ (C24 ⊕ C2n ).
Numerous other examples by Geroldinger et al. (but none for
groups of rank three or Cnr ).
Some results on D(G)
Let G = Cn1 ⊕ · · · ⊕ Cnr with ni | ni+1 . Then,
D(G) ≥ 1 +
r
X
(ni − 1) = D∗ (G).
i=1
Equality holds for (Olson, Kruyswijk, van Emde Boas, 1969)
I p-groups (group rings, later polynomial method).
I groups of rank at most 2 (inductive method, reduction to
p-groups).
In some other cases, e.g.,
I C 2 ⊕ C2n (van Emde Boas).
2
I C 2 ⊕ C3n (Bhowmik, Schlage-Puchta).
3
I C 2 ⊕ C4n and C 2 ⊕ C6n (S.).
4
6
But, not always. For example (Baayen), for odd n,
D(C24 ⊕ C2n ) > D∗ (C24 ⊕ C2n ).
Numerous other examples by Geroldinger et al. (but none for
groups of rank three or Cnr ).
Some results on D(G)
Let G = Cn1 ⊕ · · · ⊕ Cnr with ni | ni+1 . Then,
D(G) ≥ 1 +
r
X
(ni − 1) = D∗ (G).
i=1
Equality holds for (Olson, Kruyswijk, van Emde Boas, 1969)
I p-groups (group rings, later polynomial method).
I groups of rank at most 2 (inductive method, reduction to
p-groups).
In some other cases, e.g.,
I C 2 ⊕ C2n (van Emde Boas).
2
I C 2 ⊕ C3n (Bhowmik, Schlage-Puchta).
3
I C 2 ⊕ C4n and C 2 ⊕ C6n (S.).
4
6
But, not always. For example (Baayen), for odd n,
D(C24 ⊕ C2n ) > D∗ (C24 ⊕ C2n ).
Numerous other examples by Geroldinger et al. (but none for
groups of rank three or Cnr ).
The j-wise Davenport constant, Halter-Koch, 92
Dj (G) the smallest integer such that each sequence of length at
least Dj (G) has j disjoint zero-sum subsequence.
Equivalently: determine the maximum length of a sequence in
G without j disjoint zero-sum subsequence (and add 1).
Equivalently, the maximal length of a zero-sum sequence that
has not j + 1 disjoint zero-sum subsequence.
The j-wise Davenport constant, Halter-Koch, 92
Dj (G) the smallest integer such that each sequence of length at
least Dj (G) has j disjoint zero-sum subsequence.
Equivalently: determine the maximum length of a sequence in
G without j disjoint zero-sum subsequence (and add 1).
Equivalently, the maximal length of a zero-sum sequence that
has not j + 1 disjoint zero-sum subsequence.
The j-wise Davenport constant, Halter-Koch, 92
Dj (G) the smallest integer such that each sequence of length at
least Dj (G) has j disjoint zero-sum subsequence.
Equivalently: determine the maximum length of a sequence in
G without j disjoint zero-sum subsequence (and add 1).
Equivalently, the maximal length of a zero-sum sequence that
has not j + 1 disjoint zero-sum subsequence.
Motivation
Let OK be the ring of integers of some number field K , with
class group G. In OK there are approx.
x
(log log x)Dj (G)−1
log x
non-associated irreducible elements of absolute norm at most
x that are the product of at most j irreducible elements [in the
sense, cannot be written as the product of j + 1 or more irred.
elements.
Why does this constant arise?
One needs to count those elements whose image under the
transfer homomorphism is a zero-sum sequence that has not
j + 1 disjoint zero-sum subsequence.
For any nonempty sequence S over the class group of OK the
ring of integers of some number field K there are approx.
x
(log log x)|S|−1
log x
non-associated irreducible elements of absolute norm at most x
Additional motivation, Recasting the Inductive Method
Delorme, Ordaz, Quiroz showed:
Let G be a finite abelian group and H a subgroup, then
D(G) ≤ DD(H) (G/H).
Sketch of Argument: Let S = g1 . . . g` be a sequence with
` = DD(H) (G/H). Project S to G/H. By definition there exist
F1 . . . FD(H) | S such that the sum of the projection of each Fi is
0 in G/H. That is its sum is in H.
Thus,
σ(F1 ) . . . σ(FD(H) )
is a sequence over H and by definition has a zero-sum
subsequence.
Additional motivation, Recasting the Inductive Method
Delorme, Ordaz, Quiroz showed:
Let G be a finite abelian group and H a subgroup, then
D(G) ≤ DD(H) (G/H).
Sketch of Argument: Let S = g1 . . . g` be a sequence with
` = DD(H) (G/H). Project S to G/H. By definition there exist
F1 . . . FD(H) | S such that the sum of the projection of each Fi is
0 in G/H. That is its sum is in H.
Thus,
σ(F1 ) . . . σ(FD(H) )
is a sequence over H and by definition has a zero-sum
subsequence.
Additional motivation, Recasting the Inductive Method
Delorme, Ordaz, Quiroz showed:
Let G be a finite abelian group and H a subgroup, then
D(G) ≤ DD(H) (G/H).
Sketch of Argument: Let S = g1 . . . g` be a sequence with
` = DD(H) (G/H). Project S to G/H. By definition there exist
F1 . . . FD(H) | S such that the sum of the projection of each Fi is
0 in G/H. That is its sum is in H.
Thus,
σ(F1 ) . . . σ(FD(H) )
is a sequence over H and by definition has a zero-sum
subsequence.
Results on Dj (G)
Exact value:
Known for groups of rank at most two and in closely related
situations (Halter-Koch).
Yet, in contrast to the standard Davenport constant, not known
for general p-groups.
For elementary p-groups its is known (for all j) for:
I
C23 (Delorme, Ordaz, Quiroz)
I
C33 (Bhowmik, Schlage-Puchta)
I
C24 , C25 (Freeze, S.)
For specific j, in particular j = 2, known for some more C2r .
Results on Dj (G)
Exact value:
Known for groups of rank at most two and in closely related
situations (Halter-Koch).
Yet, in contrast to the standard Davenport constant, not known
for general p-groups.
For elementary p-groups its is known (for all j) for:
I
C23 (Delorme, Ordaz, Quiroz)
I
C33 (Bhowmik, Schlage-Puchta)
I
C24 , C25 (Freeze, S.)
For specific j, in particular j = 2, known for some more C2r .
Results on Dj (G)
Exact value:
Known for groups of rank at most two and in closely related
situations (Halter-Koch).
Yet, in contrast to the standard Davenport constant, not known
for general p-groups.
For elementary p-groups its is known (for all j) for:
I
C23 (Delorme, Ordaz, Quiroz)
I
C33 (Bhowmik, Schlage-Puchta)
I
C24 , C25 (Freeze, S.)
For specific j, in particular j = 2, known for some more C2r .
Results on Dj (G)
Exact value:
Known for groups of rank at most two and in closely related
situations (Halter-Koch).
Yet, in contrast to the standard Davenport constant, not known
for general p-groups.
For elementary p-groups its is known (for all j) for:
I
C23 (Delorme, Ordaz, Quiroz)
I
C33 (Bhowmik, Schlage-Puchta)
I
C24 , C25 (Freeze, S.)
For specific j, in particular j = 2, known for some more C2r .
Results on Dj (G)
Exact value:
Known for groups of rank at most two and in closely related
situations (Halter-Koch).
Yet, in contrast to the standard Davenport constant, not known
for general p-groups.
For elementary p-groups its is known (for all j) for:
I
C23 (Delorme, Ordaz, Quiroz)
I
C33 (Bhowmik, Schlage-Puchta)
I
C24 , C25 (Freeze, S.)
For specific j, in particular j = 2, known for some more C2r .
Results on Dj (G)
Exact value:
Known for groups of rank at most two and in closely related
situations (Halter-Koch).
Yet, in contrast to the standard Davenport constant, not known
for general p-groups.
For elementary p-groups its is known (for all j) for:
I
C23 (Delorme, Ordaz, Quiroz)
I
C33 (Bhowmik, Schlage-Puchta)
I
C24 , C25 (Freeze, S.)
For specific j, in particular j = 2, known for some more C2r .
Results on Dj (G)
Exact value:
Known for groups of rank at most two and in closely related
situations (Halter-Koch).
Yet, in contrast to the standard Davenport constant, not known
for general p-groups.
For elementary p-groups its is known (for all j) for:
I
C23 (Delorme, Ordaz, Quiroz)
I
C33 (Bhowmik, Schlage-Puchta)
I
C24 , C25 (Freeze, S.)
For specific j, in particular j = 2, known for some more C2r .
One more definition
Let (G, +) fin. ab. group. Let j ∈ N with j ≥ exp(G).
s≤j (G) denotes the smallest ` in N such that for each
P sequence
g1 . . . g` there exists ∅ =
6 I ⊂ {1, . . . , `} such that i∈I gi = 0
with |I| ≤ k .
η(G) = s≤exp(G) (G).
One more definition
Let (G, +) fin. ab. group. Let j ∈ N with j ≥ exp(G).
s≤j (G) denotes the smallest ` in N such that for each
P sequence
g1 . . . g` there exists ∅ =
6 I ⊂ {1, . . . , `} such that i∈I gi = 0
with |I| ≤ k .
η(G) = s≤exp(G) (G).
Some classical bounds on Dj (G)
Lower bound:
Let G = H ⊕ Cn with n = exp(G). Then,
Dj (G) ≥ j exp(G) + D(H) − 1.
Sharp for groups of rank ≤ 2, and some other cases; but this
is/should be a rare phenomenon.
Upper bounds:
Clearly, Dj (G) ≤ j D(G); this is only sharp for cyclic groups.
Dj (G) ≤ j exp(G) + max{D(G) − exp(G), η(G) − 2 exp(G)}.
(Sharp for groups of rank ≤ 2; some other cases but rarely.)
For example for G = C2r , r ≥ 3, one gets
r − 1 + 2j ≤ Dj (G) ≤ 2r − 4 + 2j.
Some classical bounds on Dj (G)
Lower bound:
Let G = H ⊕ Cn with n = exp(G). Then,
Dj (G) ≥ j exp(G) + D(H) − 1.
Sharp for groups of rank ≤ 2, and some other cases; but this
is/should be a rare phenomenon.
Upper bounds:
Clearly, Dj (G) ≤ j D(G); this is only sharp for cyclic groups.
Dj (G) ≤ j exp(G) + max{D(G) − exp(G), η(G) − 2 exp(G)}.
(Sharp for groups of rank ≤ 2; some other cases but rarely.)
For example for G = C2r , r ≥ 3, one gets
r − 1 + 2j ≤ Dj (G) ≤ 2r − 4 + 2j.
Some classical bounds on Dj (G)
Lower bound:
Let G = H ⊕ Cn with n = exp(G). Then,
Dj (G) ≥ j exp(G) + D(H) − 1.
Sharp for groups of rank ≤ 2, and some other cases; but this
is/should be a rare phenomenon.
Upper bounds:
Clearly, Dj (G) ≤ j D(G); this is only sharp for cyclic groups.
Dj (G) ≤ j exp(G) + max{D(G) − exp(G), η(G) − 2 exp(G)}.
(Sharp for groups of rank ≤ 2; some other cases but rarely.)
For example for G = C2r , r ≥ 3, one gets
r − 1 + 2j ≤ Dj (G) ≤ 2r − 4 + 2j.
Some classical bounds on Dj (G)
Lower bound:
Let G = H ⊕ Cn with n = exp(G). Then,
Dj (G) ≥ j exp(G) + D(H) − 1.
Sharp for groups of rank ≤ 2, and some other cases; but this
is/should be a rare phenomenon.
Upper bounds:
Clearly, Dj (G) ≤ j D(G); this is only sharp for cyclic groups.
Dj (G) ≤ j exp(G) + max{D(G) − exp(G), η(G) − 2 exp(G)}.
(Sharp for groups of rank ≤ 2; some other cases but rarely.)
For example for G = C2r , r ≥ 3, one gets
r − 1 + 2j ≤ Dj (G) ≤ 2r − 4 + 2j.
(Dj (G) − j exp(G))j∈N is eventually constant
Theorem (Freeze and S.)
Let G be a finite abelian group. There exists some D0 (G) such
that
Dj (G) = D0 (G) + j exp(G)
for all j ≥ (2|G|)4|G|+2 .
Remark 1: To get an explicit bound for ‘j sufficiently large’ is
the main part of the argument.
Remark 2: Define jD (G) as the smallest number for which
Dj (G) = D0 (G) + j exp(G) for all j ≥ jD (G).
We have jD (G) ≤ (2|G|)4|G|+2 .
(Dj (G) − j exp(G))j∈N is eventually constant
Theorem (Freeze and S.)
Let G be a finite abelian group. There exists some D0 (G) such
that
Dj (G) = D0 (G) + j exp(G)
for all j ≥ (2|G|)4|G|+2 .
Remark 1: To get an explicit bound for ‘j sufficiently large’ is
the main part of the argument.
Remark 2: Define jD (G) as the smallest number for which
Dj (G) = D0 (G) + j exp(G) for all j ≥ jD (G).
We have jD (G) ≤ (2|G|)4|G|+2 .
(Dj (G) − j exp(G))j∈N is eventually constant
Theorem (Freeze and S.)
Let G be a finite abelian group. There exists some D0 (G) such
that
Dj (G) = D0 (G) + j exp(G)
for all j ≥ (2|G|)4|G|+2 .
Remark 1: To get an explicit bound for ‘j sufficiently large’ is
the main part of the argument.
Remark 2: Define jD (G) as the smallest number for which
Dj (G) = D0 (G) + j exp(G) for all j ≥ jD (G).
We have jD (G) ≤ (2|G|)4|G|+2 .
(Dj (G) − j exp(G))j∈N is eventually constant
Theorem (Freeze and S.)
Let G be a finite abelian group. There exists some D0 (G) such
that
Dj (G) = D0 (G) + j exp(G)
for all j ≥ (2|G|)4|G|+2 .
Remark 1: To get an explicit bound for ‘j sufficiently large’ is
the main part of the argument.
Remark 2: Define jD (G) as the smallest number for which
Dj (G) = D0 (G) + j exp(G) for all j ≥ jD (G).
We have jD (G) ≤ (2|G|)4|G|+2 .
(Dj (G) − j exp(G))j∈N is eventually constant
Theorem (Freeze and S.)
Let G be a finite abelian group. There exists some D0 (G) such
that
Dj (G) = D0 (G) + j exp(G)
for all j ≥ (2|G|)4|G|+2 .
Remark 1: To get an explicit bound for ‘j sufficiently large’ is
the main part of the argument.
Remark 2: Define jD (G) as the smallest number for which
Dj (G) = D0 (G) + j exp(G) for all j ≥ jD (G).
We have jD (G) ≤ (2|G|)4|G|+2 .
Elementary 2-groups
We focus on elementary 2-groups.
I
rank ‘maximal’ relative to the exponent; complements
cyclic, rank two.
I
For most groups, in particular elementary p-groups with
large rank, the problem of determining η(G) is so wide
open that ‘little hope’ for Dj (G).
For example, r large,
(2.21 . . . )r η(C3r ) 3r
r 1+
(Edel; Bateman–Katz; Meshulam; context cap-sets, 3-AP.)
Elementary 2-groups
We focus on elementary 2-groups.
I
rank ‘maximal’ relative to the exponent; complements
cyclic, rank two.
I
For most groups, in particular elementary p-groups with
large rank, the problem of determining η(G) is so wide
open that ‘little hope’ for Dj (G).
For example, r large,
(2.21 . . . )r η(C3r ) 3r
r 1+
(Edel; Bateman–Katz; Meshulam; context cap-sets, 3-AP.)
Elementary 2-groups
We focus on elementary 2-groups.
I
rank ‘maximal’ relative to the exponent; complements
cyclic, rank two.
I
For most groups, in particular elementary p-groups with
large rank, the problem of determining η(G) is so wide
open that ‘little hope’ for Dj (G).
For example, r large,
(2.21 . . . )r η(C3r ) 3r
r 1+
(Edel; Bateman–Katz; Meshulam; context cap-sets, 3-AP.)
Elementary 2-groups
We focus on elementary 2-groups.
I
rank ‘maximal’ relative to the exponent; complements
cyclic, rank two.
I
For most groups, in particular elementary p-groups with
large rank, the problem of determining η(G) is so wide
open that ‘little hope’ for Dj (G).
For example, r large,
(2.21 . . . )r η(C3r ) 3r
r 1+
(Edel; Bateman–Katz; Meshulam; context cap-sets, 3-AP.)
Elementary 2-groups, large j
Theorem (Freeze and S.)
Let r ∈ N \ {1}. Then
r
r
|C2 | − 1
|C2 | − 1
r
≤ D0 (C2 ) ≤
+ |C2r |1/2
3
3
and kD (C2r ) ≤ b(2r − 1)/3c.
Elementary 2-groups, small j
Theorem (Plagne and S.)
For each sufficiently large integer r we have
1.261 r
1.500 r
1.723 r
1.934 r
2.137 r
2.333 r
2.523 r
2.709 r
2.890 r
≤ D2 (C2r )
≤ D3 (C2r )
≤ D4 (C2r )
≤ D5 (C2r )
≤ D6 (C2r )
≤ D7 (C2r )
≤ D8 (C2r )
≤ D9 (C2r )
≤ D10 (C2r )
≤
≤
≤
≤
≤
≤
≤
≤
≤
1.396 r ,
1.771 r ,
2.131 r ,
2.478 r ,
2.815 r ,
3.143 r ,
3.464 r ,
3.778 r ,
4.087 r .
For j = 2, Komlós and Katona–Srivastava; in a different context.
Elementary 2-groups, small j, II
Theorem (Plagne and S.)
When j tends to infinity, we have the following:
Dj (C2r )
Dj (C2r )
j
j
log 2
. lim inf
≤ lim sup
. 2 log 2
.
r →+∞
log j
r
r
log j
r →+∞
Link to coding theory
Let g1 . . . gn sequence in C2r . Consider gi = (ai1 , . . . , air )T with
aij ∈ CP
2.
Then i∈I gi = 0 if and only if
[g1 | · · · | gn ]xI = 0
where xIT = (xIj ) ∈ C2n with xIj equal 1 if j ∈ I and 0 otherwise.
Thus, g1 . . . gn has a zero-sum subsequence of length at most
d if and only if the minimal distance of the code with parity
check matrix [g1 | · · · | gn ] has minimal distance at most d.
Link to coding theory
Let g1 . . . gn sequence in C2r . Consider gi = (ai1 , . . . , air )T with
aij ∈ CP
2.
Then i∈I gi = 0 if and only if
[g1 | · · · | gn ]xI = 0
where xIT = (xIj ) ∈ C2n with xIj equal 1 if j ∈ I and 0 otherwise.
Thus, g1 . . . gn has a zero-sum subsequence of length at most
d if and only if the minimal distance of the code with parity
check matrix [g1 | · · · | gn ] has minimal distance at most d.
Link to coding theory
Let g1 . . . gn sequence in C2r . Consider gi = (ai1 , . . . , air )T with
aij ∈ CP
2.
Then i∈I gi = 0 if and only if
[g1 | · · · | gn ]xI = 0
where xIT = (xIj ) ∈ C2n with xIj equal 1 if j ∈ I and 0 otherwise.
Thus, g1 . . . gn has a zero-sum subsequence of length at most
d if and only if the minimal distance of the code with parity
check matrix [g1 | · · · | gn ] has minimal distance at most d.
Intersecting codes
A code is called intersecting if each two non-zero codewords do
not have disjoint support. (Studied by Katona, Miklós,
Cohen–Lempel,...)
The following are (essentially) equivalent [Cohen–Zémor]:
I
Determine for which n, k intersecting [n, k ]-codes exist.
I
Determine D2 (C2r ).
Intersecting codes
A code is called intersecting if each two non-zero codewords do
not have disjoint support. (Studied by Katona, Miklós,
Cohen–Lempel,...)
The following are (essentially) equivalent [Cohen–Zémor]:
I
Determine for which n, k intersecting [n, k ]-codes exist.
I
Determine D2 (C2r ).
Intersecting codes
A code is called intersecting if each two non-zero codewords do
not have disjoint support. (Studied by Katona, Miklós,
Cohen–Lempel,...)
The following are (essentially) equivalent [Cohen–Zémor]:
I
Determine for which n, k intersecting [n, k ]-codes exist.
I
Determine D2 (C2r ).
Argument for the upper bounds
Delorme, Ordaz, and Quiroz:
Dj+1 (G) ≤ min max{Dj (G) + i, s≤i (G) − 1}.
i∈N
Need/want knowledge on s≤i (C2r ); then apply repeatedly.
Argument for the upper bounds
Delorme, Ordaz, and Quiroz:
Dj+1 (G) ≤ min max{Dj (G) + i, s≤i (G) − 1}.
i∈N
Need/want knowledge on s≤i (C2r ); then apply repeatedly.
Some ad-hoc terminology
Let f : [0, 1] → [0, 1] (non-increasing, continuous, and) each
[n, k , d] code (binary linear) satisfies
k
d
≤f
.
n
n
I.o.w., the functions in the upper bounds of the rate of a code by
a function of its normalized minimal distance. Call it
“upper-bounding function”; and “asypmtotically upper-bounding
function” if holds for all large n.
E.g. Hamming bound:
δ
.
f (δ) = 1 − h
2
with
h(u) = −u log2 u − (1 − u) log2 (1 − u)
binary entropy.
Some ad-hoc terminology
Let f : [0, 1] → [0, 1] (non-increasing, continuous, and) each
[n, k , d] code (binary linear) satisfies
k
d
≤f
.
n
n
I.o.w., the functions in the upper bounds of the rate of a code by
a function of its normalized minimal distance. Call it
“upper-bounding function”; and “asypmtotically upper-bounding
function” if holds for all large n.
E.g. Hamming bound:
δ
f (δ) = 1 − h
.
2
with
h(u) = −u log2 u − (1 − u) log2 (1 − u)
binary entropy.
Key lemma
Lemma
Let f be an [asymptotic] upper-bounding function. Let d, n, and
r be three positive integers [n sufficiently large] satisfying
2 ≤ d ≤ n − 1 and
d +1
n−r
>f
,
n
n
then
s≤d (C2r ) ≤ n.
Upper bounds, summary
I
Use DOQ to reduce to s≤i (C2r ).
I
Reduce s≤i (C2r ) to “bounds on codes.”
I
Use bounds from coding theory (small j, McEliece,
Rodemich, Rumsey, and Welch; asymt. Hamming)
I
Perform some computations and assemble the pieces.
Upper bounds, summary
I
Use DOQ to reduce to s≤i (C2r ).
I
Reduce s≤i (C2r ) to “bounds on codes.”
I
Use bounds from coding theory (small j, McEliece,
Rodemich, Rumsey, and Welch; asymt. Hamming)
I
Perform some computations and assemble the pieces.
Upper bounds, summary
I
Use DOQ to reduce to s≤i (C2r ).
I
Reduce s≤i (C2r ) to “bounds on codes.”
I
Use bounds from coding theory (small j, McEliece,
Rodemich, Rumsey, and Welch; asymt. Hamming)
I
Perform some computations and assemble the pieces.
Upper bounds, summary
I
Use DOQ to reduce to s≤i (C2r ).
I
Reduce s≤i (C2r ) to “bounds on codes.”
I
Use bounds from coding theory (small j, McEliece,
Rodemich, Rumsey, and Welch; asymt. Hamming)
I
Perform some computations and assemble the pieces.
Lower bounds
Let j be a positive integer. Then
Dj (C2r ) ≥ log 2
j
r
log(j + 1)
as r tends to infinity.
Proved via a counting argument similar to argument of
Cohen–Lempel for intersecting codes, j = 2.
Lower bounds
Let j be a positive integer. Then
Dj (C2r ) ≥ log 2
j
r
log(j + 1)
as r tends to infinity.
Proved via a counting argument similar to argument of
Cohen–Lempel for intersecting codes, j = 2.
True value?
Extrapolating a Conjecture of Cohen–Lempel:
For any positive integer j,
Dj (C2r )
j
lim
∼ log 2
.
r →+∞
r
log j
That is the lower bound.
Generalizations, work in progress (with Ordaz et al.)
While the coding theoretic approach does not generalize well
(at all?) to Cpr ,
it can be used to investigate ±-weighted zero-sum constants for
C3r ,
or ’fully-weighted’ zero-sum constants for Cpr (p arbitrary).
We have obtained analogous results...
...but the details need to be polished/rechecked.
Generalizations, work in progress (with Ordaz et al.)
While the coding theoretic approach does not generalize well
(at all?) to Cpr ,
it can be used to investigate ±-weighted zero-sum constants for
C3r ,
or ’fully-weighted’ zero-sum constants for Cpr (p arbitrary).
We have obtained analogous results...
...but the details need to be polished/rechecked.
Generalizations, work in progress (with Ordaz et al.)
While the coding theoretic approach does not generalize well
(at all?) to Cpr ,
it can be used to investigate ±-weighted zero-sum constants for
C3r ,
or ’fully-weighted’ zero-sum constants for Cpr (p arbitrary).
We have obtained analogous results...
...but the details need to be polished/rechecked.
Generalizations, work in progress (with Ordaz et al.)
While the coding theoretic approach does not generalize well
(at all?) to Cpr ,
it can be used to investigate ±-weighted zero-sum constants for
C3r ,
or ’fully-weighted’ zero-sum constants for Cpr (p arbitrary).
We have obtained analogous results...
...but the details need to be polished/rechecked.
Generalizations, work in progress (with Ordaz et al.)
While the coding theoretic approach does not generalize well
(at all?) to Cpr ,
it can be used to investigate ±-weighted zero-sum constants for
C3r ,
or ’fully-weighted’ zero-sum constants for Cpr (p arbitrary).
We have obtained analogous results...
...but the details need to be polished/rechecked.