Lattice packings: an upper bound on the number of perfect lattices - - PowerPoint PPT Presentation

Lattice packings: an upper bound

• n the number of perfect lattices

Wessel van Woerden, CWI, Amsterdam.

1 | 23 Sphere Packing Problem

1 | 23 Sphere Packing Problem

1 | 23 Sphere Packing Problem

• Only solved in dimensions 2, 3, 8 and 24...

2 | 23 Lattice Packing Problem

b1 b2 b1 + b2

2 | 23 Lattice Packing Problem

b1 b2 b1 + b2

• Solved in dimensions ≤ 8 and 24.

3 | 23 Solution space

• Cone of positive definite quadratic forms:

4 | 23 Ryshkov Polyhedron

• Spheres of radius at least 1:

4 | 23 Ryshkov Polyhedron

• Spheres of radius at least 1:
• Concave minimization problem =

⇒ optima at vertices.

4 | 23 Ryshkov Polyhedron

• Spheres of radius at least 1:
• Concave minimization problem =

⇒ optima at vertices.

• Finite number of non-similar vertices.

4 | 23 Ryshkov Polyhedron

• Spheres of radius at least 1:
• Concave minimization problem =

⇒ optima at vertices.

• Finite number of non-similar vertices. ← how many?

5 | 23 Voronoi’s Algorithm

• How to solve the lattice packing problem in a fixed dimension:
• Enumerate all non-similar vertices.

5 | 23 Voronoi’s Algorithm

• How to solve the lattice packing problem in a fixed dimension:
• Enumerate all non-similar vertices.
• Pick the best one.

6 | 23 Space of quadratic forms

• Let Sd ⊂ Rd×d be the space of real symmetric matrices.

6 | 23 Space of quadratic forms

• Let Sd ⊂ Rd×d be the space of real symmetric matrices.
• Dimension n := 1

2d(d + 1).

6 | 23 Space of quadratic forms

• Let Sd ⊂ Rd×d be the space of real symmetric matrices.
• Dimension n := 1

2d(d + 1).

• We define the following inner product on Sd:

A, B := Tr(AtB) =

• i,j

AijBij

6 | 23 Space of quadratic forms

• Let Sd ⊂ Rd×d be the space of real symmetric matrices.
• Dimension n := 1

2d(d + 1).

• We define the following inner product on Sd:

A, B := Tr(AtB) =

• i,j

AijBij

• Q ∈ Sd defines a quadratic form by

Q[x] := xtQx = Q, xxt ∀x ∈ Rd

6 | 23 Space of quadratic forms

• Let Sd ⊂ Rd×d be the space of real symmetric matrices.
• Dimension n := 1

2d(d + 1).

• We define the following inner product on Sd:

A, B := Tr(AtB) =

• i,j

AijBij

• Q ∈ Sd defines a quadratic form by

Q[x] := xtQx = Q, xxt ∀x ∈ Rd

• For a positive definite quadratic form (PQF) Q ∈ Sd

>0:

λ(Q) := min

x∈Zd−{0} Q[x]

Min (Q) := {x ∈ Zd : Q[x] = λ(Q)}

7 | 23 Hermite Constant

• Lattice L = BZd =

⇒ PQF Q = BtB ∈ Sd

>0.

7 | 23 Hermite Constant

• Lattice L = BZd =

⇒ PQF Q = BtB ∈ Sd

>0.

7 | 23 Hermite Constant

• Lattice L = BZd =

⇒ PQF Q = BtB ∈ Sd

>0.

∼ det(Q)1/2 ∼ λ(Q)d/2

7 | 23 Hermite Constant

• Lattice L = BZd =

⇒ PQF Q = BtB ∈ Sd

>0.

∼ det(Q)1/2 ∼ λ(Q)d/2

• Hermite invariant:

γ(Q) = λ(Q) (det Q)1/d ∼ density(L)2/d

7 | 23 Hermite Constant

• Lattice L = BZd =

⇒ PQF Q = BtB ∈ Sd

>0.

∼ det(Q)1/2 ∼ λ(Q)d/2

• Hermite invariant:

γ(Q) = λ(Q) (det Q)1/d ∼ density(L)2/d

• Lattice packing problem ⇔ determine Hermite’s constant:

Hd := sup

Q∈Sd

>0

γ(Q)

8 | 23 Ryshkov Polyhedra

• For λ > 0 we define the Ryshkov Polyhedron

Pλ = {Q ∈ Sd

>0 : λ(Q) ≥ λ}

8 | 23 Ryshkov Polyhedra

• For λ > 0 we define the Ryshkov Polyhedron

Pλ =

• x∈Zd\{0}

{Q ∈ Sd : Q, xxt ≥ λ} ⊂ Sd

>0

8 | 23 Ryshkov Polyhedra

• For λ > 0 we define the Ryshkov Polyhedron

Pλ =

• x∈Zd\{0}

{Q ∈ Sd : Q, xxt ≥ λ} ⊂ Sd

>0

• Facets correspond to x ∈ Zd \ {0}.

8 | 23 Ryshkov Polyhedra

• For λ > 0 we define the Ryshkov Polyhedron

Pλ = {Q ∈ Sd

>0 : λ(Q) ≥ λ}

• We have

Hd = λ inf

Q∈Pλ det(Q)1/d

8 | 23 Ryshkov Polyhedra

• For λ > 0 we define the Ryshkov Polyhedron

Pλ = {Q ∈ Sd

>0 : λ(Q) ≥ λ}

• We have

Hd = λ inf

Q∈Pλ det(Q)1/d

• Minkowski: det(Q)1/d is (strictly) concave on Sd

>0

= ⇒ Local optima at vertices of Pλ.

9 | 23 Perfect forms

• Q is perfect ⇔ Q is a vertex of Pλ(Q).

9 | 23 Perfect forms

• Q is perfect ⇔ Q is a vertex of Pλ(Q).
• Note that |Min Q| ≥ 2n = d(d + 1)

10 | 23 Similarity

• B and BU generate the same lattice for U ∈ GLd(Z).

10 | 23 Similarity

• B and BU generate the same lattice for U ∈ GLd(Z).
• Arithmetically equivalence: ∃U s.t. Q′ = UtQU.

10 | 23 Similarity

• B and BU generate the same lattice for U ∈ GLd(Z).
• Arithmetically equivalence: ∃U s.t. Q′ = UtQU.
• Note that λ1(UtQU) = λ1(Q) and det(UtQU) = det(Q).

10 | 23 Similarity

• B and BU generate the same lattice for U ∈ GLd(Z).
• Arithmetically equivalence: ∃U s.t. Q′ = UtQU.
• Note that λ1(UtQU) = λ1(Q) and det(UtQU) = det(Q).
• Similarity: Arithmetical equivalence up to scaling.

Perfect Forms: how many?

11 | 23 Number of perfect forms

• The exact set of perfect forms is known up to dimension 8.
• For d ≥ 6 Voronoi’s Algorithm was used.

d # non-similar Perfect forms 2 1 (Lagrange, 1773) 3 1 (Gauss, 1840) 4 2 (Korkine & Zolotarev, 1877) 5 3 (Korkine & Zolotarev, 1877) 6 7 (Barnes, 1957) 7 33 (Jaquet, 1993) 8 10916 (DSV, 2005) 9 ≥ 500.000 (DSV, 2005) ≥ 23.000.000 (vW, 2018)

12 | 23 An improved upper bound

• pd := number of non-similar d-dimensional perfect forms.

12 | 23 An improved upper bound

• pd := number of non-similar d-dimensional perfect forms.
• Known bounds for pd.

pd < eO(d4 log(d)) (C. Soul´ e, 1998) eΩ(d) < pd < eO(d3 log(d)) (R. Bacher, 2017)

12 | 23 An improved upper bound

• pd := number of non-similar d-dimensional perfect forms.
• Known bounds for pd.

pd < eO(d4 log(d)) (C. Soul´ e, 1998) eΩ(d) < pd < eO(d3 log(d)) (R. Bacher, 2017)

Theorem (This talk)

pd < eO(d2 log(d))

13 | 23 Outer Normal Cones

polyhedron inside cone = ⇒ subdivision of cone

1 2 3 4 1 4 2 3

13 | 23 Inner Normal Cones

polyhedron inside cone = ⇒ subdivision of cone

1 2 3 4 4 1 3 2

14 | 23 Subdivision for d = 2

Figure: Subdivision by normal cones of Ryshkov Polyhedron.

15 | 23 Voronoi Domain

Definition

For a PQF Q ∈ Sd

>0 its Voronoi Domain V(Q) is

V(Q) := cone({xxt : x ∈ Min Q}) ⊂ Sd

≥0.

15 | 23 Voronoi Domain

Definition

For a PQF Q ∈ Sd

>0 its Voronoi Domain V(Q) is

V(Q) := cone({xxt : x ∈ Min Q}) ⊂ Sd

≥0.

• Q is perfect ⇔ V(Q) is full dimensional.

16 | 23 Proof strategy

(0, 1) (1, 0) (1, 1) (−1, 1) (1, 2) (−1, 2) (2, 1) (−2, 1) (1, 3) (−1, 3) (3, 1) (−3, 1) (2, 3) (−2, 3) (3, 2) (−3, 1)

16 | 23 Proof strategy

(0, 1) (1, 0) (1, 1) (−1, 1) (1, 2) (−1, 2) (2, 1) (−2, 1) (1, 3) (−1, 3) (3, 1) (−3, 1) (2, 3) (−2, 3) (3, 2) (−3, 1)

Can’t fit many large Voronoi domains.

16 | 23 Proof strategy

(0, 1) (1, 0) (1, 1) (−1, 1) (1, 2) (−1, 2) (2, 1) (−2, 1) (1, 3) (−1, 3) (3, 1) (−3, 1) (2, 3) (−2, 3) (3, 2) (−3, 1)

Can’t fit many large Voronoi domains. Each similariy-class has at least one large representative.

16 | 23 Proof strategy

(0, 1) (1, 0) (1, 1) (−1, 1) (1, 2) (−1, 2) (2, 1) (−2, 1) (1, 3) (−1, 3) (3, 1) (−3, 1) (2, 3) (−2, 3) (3, 2) (−3, 1)

Can’t fit many large Voronoi domains. Each similariy-class has at least one large representative. = ⇒ Can’t have many similarity-classes.

17 | 23 Volumetric argument

• Find a complete set of representatives Pd such that:

Vol

• Sd

≥0

• ≤ ud

Vol (V(Q)) ≥ ℓd ∀Q ∈ Pd

17 | 23 Volumetric argument

• Find a complete set of representatives Pd such that:

Vol

• Sd

≥0

• ≤ ud

Vol (V(Q)) ≥ ℓd ∀Q ∈ Pd

• Then pd = |Pd| ≤ ud

ℓd .

17 | 23 Volumetric argument

• Find a complete set of representatives Pd such that:

Vol

• Sd

≥0

• ≤ ud

= o(1) Vol (V(Q)) ≥ ℓd ∀Q ∈ Pd

• Then pd = |Pd| ≤ ud

ℓd .

• To quantify the volume we restrict to the half space

Td := {Q ∈ Sd : Tr(Q) = Q, Id ≤ 1}.

18 | 23 Volume simplex

a1 a3 a2 n-dimensional simplex: Volume = 1

n! · |det(ai, aj)i,j|1/2

19 | 23 Volume Voronoi domain

• Tr(xxt) = xtx.

19 | 23 Volume Voronoi domain

• Tr(xxt) = xtx.
• Can look at subcone: w.l.o.g. Min Q = {±x1, . . . , ±xn}.

x1xt

1

xt

1x1

x3xt

3

xt

3x3

x2xt

2

xt

2x2

V(Q) ∩ Td

20 | 23 Lower bound

We get Vol(V(Q) ∩ Td) = 1 n! ·

• det
• xixt

i

xt

i xi

, xjxt

j

xt

j xj

• i,j∈[n]
• 1/2

20 | 23 Lower bound

We get Vol(V(Q) ∩ Td) = 1 n! ·

• det
• xixt

i

xt

i xi

, xjxt

j

xt

j xj

• i,j∈[n]
• 1/2

= 1 n! ·

n

• i=1

1 xt

i xi

• ·
• det
• xixt

i , xjxt j

• i,j∈[n]
• 1/2

20 | 23 Lower bound

We get Vol(V(Q) ∩ Td) = 1 n! ·

• det
• xixt

i

xt

i xi

, xjxt

j

xt

j xj

• i,j∈[n]
• 1/2

= 1 n! ·

n

• i=1

1 xt

i xi

• ·
• det
• xixt

i , xjxt j

• i,j∈[n]
• 1/2

≥ 1 n! ·

n

• i=1

1 xt

i xi

• ← Uses integrality of the xi

20 | 23 Lower bound

We get Vol(V(Q) ∩ Td) = 1 n! ·

• det
• xixt

i

xt

i xi

, xjxt

j

xt

j xj

• i,j∈[n]
• 1/2

= 1 n! ·

n

• i=1

1 xt

i xi

• ·
• det
• xixt

i , xjxt j

• i,j∈[n]
• 1/2

≥ 1 n! ·

n

• i=1

1 xt

i xi

• ← Uses integrality of the xi

≥ℓd? We need to upper bound all xt

i xi.

21 | 23 Short minimal vectors

Lemma

Let PQF Q ∈ Sd

>0. Then there exists a Q′ arithmetically

equivalent to Q such that xtx = O(d4) ∀x ∈ Min Q′

21 | 23 Short minimal vectors

Lemma

Let PQF Q ∈ Sd

>0. Then there exists a Q′ arithmetically

equivalent to Q such that xtx = O(d4) ∀x ∈ Min Q′

• Proof: transference and dual lattice reduction.

21 | 23 Short minimal vectors

Lemma

Let PQF Q ∈ Sd

>0. Then there exists a Q′ arithmetically

equivalent to Q such that xtx = O(d4) ∀x ∈ Min Q′

• Proof: transference and dual lattice reduction.

Vol(V(Q) ∩ Td) ≥ 1 n! ·

n

• i=1

1 xt

i xi

• ≥ 1

n! ·

• 1

O (d4)

n

=: ℓd

22 | 23 Conclusion

Remind that n = 1

2d(d + 1). To conclude:

pd = |Pd| ≤ ud ℓd = o(1) · n! · O(d4)n

22 | 23 Conclusion

Remind that n = 1

2d(d + 1). To conclude:

pd = |Pd| ≤ ud ℓd = o(1) · n! · O(d4)n = eO(d2 log(d))

22 | 23 Conclusion

Remind that n = 1

2d(d + 1). To conclude:

pd = |Pd| ≤ ud ℓd = o(1) · n! · O(d4)n = eO(d2 log(d))

Thank you!

23 | 23 Citations

• C. Soul´

e, Perfect forms and the Vandiver conjecture, Journal fur die Reine und Angewandte Mathematik 517 (1999) 209–222.

• R. Bacher, On the number of perfect lattices, Journal de

Th´ eorie des Nombres de Bordeaux 30 (3) (2018) 917–945.

• J. Martinet, Perfect Lattices in Euclidean Spaces, Grundlehren

der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2002.

• A. Schurmann, Computational Geometry of Positive Definite

Quadratic Forms : Polyhedral Reduction Theories, Algorithms, and Applications, American Mathematical Society, 2009.