weyl’s law for heisenberg manifolds chung, khosravi...

Weyl’s Law forHeisenbergManifolds

Chung, Khosravi,Petridis, Toth

Outline

Weyl’s Law for Heisenberg Manifolds

Derrick Chung1 Mahta Khosravi2 Yiannis Petridis1

John Toth3

1The Graduate Center and Lehman College, City University of New York2Institute for Advanced Study 3McGill University

November 10, 2006

The setting

The Heisenberg group

Heisenberg manifolds

Why do we care?

The spectrum ofHeisenberg manifolds

Classicallattice-countingproblems

Results andConjectures

Weyl’s Law

Exponent pairs

Evidence

Average over metrics

Higher dimensions

Hint of proof of Th. 1

Open problems

Outline

The settingThe Heisenberg groupHeisenberg manifoldsWhy do we care?The spectrum of Heisenberg manifoldsClassical lattice-counting problems

Results and ConjecturesWeyl’s LawExponent pairsEvidenceAverage over metricsNumericsHigher dimensionsHint of proof of Th. 1

Open problems

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Heisenberg algebra:

hn = 〈X1, . . . ,Xn,Y1, . . . ,Yn,Z 〉

[Xi ,Xj ] = [Yi ,Yj ] = [Xi ,Z ] = [Yi ,Z ] = 0

[Xi ,Yj ] = δijZ

Heisenberg Group:

0 Inty

, x, y ∈ Rn, z ∈ R

X (x, y, z) =

0 x z0 0 ty0 0 0

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

hn = {X (x, y, z)} ⊂ gl(n + 2,R)

{X (x, y, 0), x, y ∈ Rn} ≡ R2n

Center, derived subalgebra: zn = {X (0, 0, z)}

hn = R2n + zn

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Heisenberg Manifolds

(Γ \ Hn, g)

Γ uniform discrete, g left-invariant metric

S1 ↪→ Γ \ Hn

↓T 2n

Circle bundle over a torusTake r ∈ Zn

+, rj |rj+1. Define

0 Inty

, xi ∈ riZ, y ∈ Zn, z ∈ Z

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Theorem (Gordon-Wilson)

(a) ∃r:(Γ \ Hn, g) ∼ (Γr \ Hn, g)

(b) ∃φ ∈ Inn(Hn) : hn = R2n ⊕ zn, rel. φ∗(g)

φ∗(g) =

[h2n×2n 0

0 g2n+1

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Why do we care?

1. Hn model for CR manifolds (Folland, Stein, Beals,Greiner)

2. Integrable: Butler, J. Geom. Phys. 36(2000)One integral is NOT analytic functionTaimanov: Analytic integrals constrain π1(M)

3. Isospectral manifolds: Gordon, Wilson, Gornet, Pescen = 1, (Γ \ H1, g) determined by its spectrum amongHeisenbergn > 1, if r1r2 · · · rn = r ′1r

′2 · · · r ′n, continuous families

Spec(Γ \ Hn, gt) = Spec(Γ′ \ Hn, g′t)

Almost Inner Automorphisms exist in abundance.

4. (H1, g) is a model geometry in classification of3-manifolds

5. Fourier coefficients of automorphic forms are∫Γ∩N\N φ(ng) dn where Γ ∩ N \ N is Heisenberg

manifold.

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Why do we care?

manifold.

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Why do we care?

manifold.

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Why do we care?

manifold.

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Why do we care?

manifold.

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Why do we care?

manifold.

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

The spectrum of Heisenberg manifolds

Deninger-Singhof, Colin de Verdiere, Gordon-WilsonLet Lr = {X (x, y, 0), xi ∈ riZ, y ∈ Zn}

[0 In−In 0

]±id2

j be the eigenvalues of h−1J

The Spectrum

1. Type I: Spec(Lr \ R2n, h)

2. Type II: µ(y , t1, t2, . . . , tn) =4π2y2

g2n+1+

n∑j=1

2πyd2j (2tj +

1), y ∈ N, ti ∈ Z+,with multiplicity 2ynr1r2 · · · rn.

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

[0 In−In 0

]±id2

The Spectrum

2. Type II: µ(y , t1, t2, . . . , tn) =4π2y2

g2n+1+

n∑j=1

2πyd2j (2tj +

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

[0 In−In 0

]±id2

The Spectrum

2. Type II: µ(y , t1, t2, . . . , tn) =4π2y2

g2n+1+

n∑j=1

2πyd2j (2tj +

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

[0 In−In 0

]±id2

The Spectrum

2. Type II: µ(y , t1, t2, . . . , tn) =4π2y2

g2n+1+

n∑j=1

2πyd2j (2tj +

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?

Example

I On H1

[I2 00 2π

]I 2π factors:

2πµ(y , t) =

(y2 + y(2t + 1)

)= y(y + 2t + 1) = yx

with x > y , x 6≡ y(mod 2)

I Leads to lattice-point counting with weight y belowhyperbola xy = λ and line x = yUse Z2 and L = {(x , y) ∈ Z2, x ≡ y(mod 2)}

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Example

I On H1

[I2 00 2π

]I 2π factors:

2πµ(y , t) =

(y2 + y(2t + 1)

)= y(y + 2t + 1) = yx

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Example

I On H1

[I2 00 2π

I 2π factors:

2πµ(y , t) =

(y2 + y(2t + 1)

)= y(y + 2t + 1) = yx

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Example

I On H1

[I2 00 2π

]I 2π factors:

2πµ(y , t) =

(y2 + y(2t + 1)

)= y(y + 2t + 1) = yx

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Example

I On H1

[I2 00 2π

]I 2π factors:

2πµ(y , t) =

(y2 + y(2t + 1)

)= y(y + 2t + 1) = yx

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems2 4 6 8 10

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems2 4 6 8 10

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Gauß circle problem

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Gauß circle problem

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Gauß circle problem and Hardy’s conjecture

I N(λ) = #{(x , y) ∈ Z2, x2 + y2 ≤ λ}

I N(λ) = πλ+ R(λ)

I R(λ) = O(λ1/2)

Hardy conjecture

R(λ) = O(λ1/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

I N(λ) = #{(x , y) ∈ Z2, x2 + y2 ≤ λ}I N(λ) = πλ+ R(λ)

I R(λ) = O(λ1/2)

Hardy conjecture

R(λ) = O(λ1/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

I R(λ) = O(λ1/2)

Hardy conjecture

R(λ) = O(λ1/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

I R(λ) = O(λ1/2)

Hardy conjecture

R(λ) = O(λ1/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Dirichlet divisor problem

I N(λ) = #{(x , y) ∈N2, xy ≤ λ}

I N(λ) =λ log λ+(2γ−1)λ+∆(λ)

I ∆(λ) = O(λ1/2)

Conjecture

∆(λ) = O(λ1/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

I N(λ) = #{(x , y) ∈N2, xy ≤ λ}

I N(λ) =λ log λ+(2γ−1)λ+∆(λ)

I ∆(λ) = O(λ1/2)

Conjecture

∆(λ) = O(λ1/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

I N(λ) = #{(x , y) ∈N2, xy ≤ λ}

I N(λ) =λ log λ+(2γ−1)λ+∆(λ)

I ∆(λ) = O(λ1/2)

Conjecture

∆(λ) = O(λ1/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

I N(λ) = #{(x , y) ∈N2, xy ≤ λ}

I N(λ) =λ log λ+(2γ−1)λ+∆(λ)

I ∆(λ) = O(λ1/2)

Conjecture

∆(λ) = O(λ1/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

DefinitionSpectral counting function

N(λ) = #{λj ≤ λ}

Weyl’s law, Hormander’s Theorem

N(λ) = cnvol(M)λn/2 + R(λ)

withR(λ) = O(λ(n−1)/2)

RemarkIn 3-dim gives R(λ) = O(λ)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

N(λ) = #{λj ≤ λ}

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

N(λ) = #{λj ≤ λ}

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Theorem (Chung, P., Toth)

For every left-invariant metric g on H1

R(λ) = O(λ34/41)

Conjecture 1:

R(λ) = O(λ3/4+ε)

RemarkFor Z3 \ R3, conj.: R(λ) = O(λ1/2+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

R(λ) = O(λ34/41)

Conjecture 1:

R(λ) = O(λ3/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

R(λ) = O(λ34/41)

Conjecture 1:

R(λ) = O(λ3/4+ε)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Theorem (Chung, P., Toth))

If (k, l) is an exponent pair then R(λ) = O(λ(l+2k+1)/(2k+2))

DefinitionLet 0 ≤ k ≤ 1/2 ≤ l ≤ 1. We call (k, l) an exponent pair if∀s > 0 ∃P(k, l , s) ∀N > 0 ∀t > 0 and ∀f (x) withf ′(x) ∼ tx−s , f ′′(x) ∼ −stx−s−1, . . .

f (P)(x) ∼ (−1)P+1s(s + 1) · · · (s + P − 2)tx−s−P+1

we have ∑N≤n≤2N

e(f (n)) � (tN−s)kN l + t−1Ns

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Theorem (Chung, P., Toth))

If (k, l) is an exponent pair then R(λ) = O(λ(l+2k+1)/(2k+2))

DefinitionLet 0 ≤ k ≤ 1/2 ≤ l ≤ 1. We call (k, l) an exponent pair if∀s > 0 ∃P(k, l , s) ∀N > 0 ∀t > 0 and ∀f (x) withf ′(x) ∼ tx−s , f ′′(x) ∼ −stx−s−1, . . .

f (P)(x) ∼ (−1)P+1s(s + 1) · · · (s + P − 2)tx−s−P+1

we have ∑N≤n≤2N

e(f (n)) � (tN−s)kN l + t−1Ns

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Conjecture (Montgomery)

∀ε, (ε, 1/2 + ε) is an exponent pair.

I Implies Lindelof H, Hardy’s conjecture

I Implies Conjecture 1 for Heisenberg

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Evidence

Theorem (Khosravi, Toth)

Cramer’s formula

limT→∞

0R(λ)2 dλ = c > 0

Compare with Cramer (1922)

I For Gauss circle problem

limT→∞

0R(λ)2 dλ =

I For Dirichlet divisor problem

limT→∞

0∆(λ)2 dλ =

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Evidence

Cramer’s formula

limT→∞

0R(λ)2 dλ = c > 0

limT→∞

0R(λ)2 dλ =

limT→∞

0∆(λ)2 dλ =

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Evidence

Cramer’s formula

limT→∞

0R(λ)2 dλ = c > 0

limT→∞

0R(λ)2 dλ =

limT→∞

0∆(λ)2 dλ =

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Evidence

Cramer’s formula

limT→∞

0R(λ)2 dλ = c > 0

limT→∞

0R(λ)2 dλ =

limT→∞

0∆(λ)2 dλ =

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Theorem (P., Toth)

For a local perturbation gu, u ∈ I of g0∫IR(λ, u)2 du = O(λ3/2)

Kendall (1948)

Shifts of Z2

0R(λ, α, β)2 dα dβ = O(λ1/2)

Theorem (P., Toth)

For a local perturbation Lu of the standard lattice Z2∫IR(λ, u)2 du = O(λ1/2)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Theorem (P., Toth)

Kendall (1948)

Shifts of Z2

0R(λ, α, β)2 dα dβ = O(λ1/2)

Theorem (P., Toth)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Theorem (P., Toth)

Kendall (1948)

Shifts of Z2

0R(λ, α, β)2 dα dβ = O(λ1/2)

Theorem (P., Toth)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Lower bounds

Theorem (P., Toth)

For fixed g1

∫ 2T

T|R(λ)| dλ� T 3/4

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.8

−0.6

−0.4

−0.2

Error Term for N(t)

Figure: The relative error for the standard lattice E (t)/t3/4

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−8000

−6000

−4000

−2000

Error Term for N(t)

Figure: The absolute error for the standard lattice

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.8

−0.6

−0.4

−0.2

Error Term for NL(t)

Figure: The relative error for the lattice L

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Error term for the Dirichlet Divisor Problem

Figure: The relative error in the Dirichlet divisor problem

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Error term for the Dirichlet Divisor Problem

Figure: The absolute error in the Dirichlet divisor problem

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

6Error Term for Gauss Circle Problem

Figure: The relative error term in Gauss’ circle problem

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

8000Histogram for N(t) (normalized)

Figure: The Histogram for the standard lattice

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

Histogram for NL(t) (normalized)

Figure: Histogram for the Lattice L

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

−4 −3 −2 −1 0 1 2 3 4 5 60

8000Histogram for Dirichlet Divisor Problem (normalized)

Figure: Histogram for Dirichlet’s Divisor problem

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

−8 −6 −4 −2 0 2 4 60

8000Histogram for Gauss Circle Problem (normalized)

Figure: Histogram for Gauss’ circle problem

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Higher dimensions

Hormander: R(λ) = O(λn) on Hn

If d2i /d

2j are all rational, we call g rational.

Theorem (Khosravi, P.)

I Rational case: R(λ) = O(λn−7/41)

I Irrational case: R(λ) = O(λn−1/4+ε) for generic g

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Higher dimensions

If d2i /d

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Higher dimensions

If d2i /d

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Higher dimensions

If d2i /d

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Conjecture 2 (Khosravi, P.)

n > 1 and g rational:

R(λ) = O(λn−1/4+ε)

I Phase is linear in n − 1 parameters.

I A generic irrational θ satisfies diophantine condition

||jθ|| � 1

j log2 j

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

R(λ) = O(λn−1/4+ε)

||jθ|| � 1

j log2 j

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

R(λ) = O(λn−1/4+ε)

||jθ|| � 1

j log2 j

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Need two term asymptotics for lattice point countingNeed to see cancellation with NI (λ) ∼ cλ for Type I (torus)eigenvaluesFor g0, x , y ∈ N∑

xy≤λ

y =∑

y≤√

√λ<x≤λ/y

1 =∑

y≤√

y([λ/y ]− [√λ])

y≤√

y(λ/y −√λ− ψ(λ/y) + ψ(

√λ))

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

where ψ(u) = u − [u]− 1/2

ψ(u) = −∑n 6=0

e2πinu

Use Euler summation∑n≤X

n =X 2

2− ψ(X )X + O(1)

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Distribution of λ−3/4R(λ)

QuestionIs λ−3/4R(λ) in B2 Besicovitch class?

Theorem (Khosravi 2006)∫ T

0R(x)3 dx ∼ cT 13/4, T →∞, c > 0

Heath-Brown (1992)

∃f (a) for Gauss and g(a) for Dirichlet divisor:

Xmeasure{x ∈ [1,X ], x−1/4R(x) ∈ I} →

∫If (a) da

Xmeasure{x ∈ [1,X ], x−1/4∆(x) ∈ I} →

∫Ig(a) da

The setting

Why do we care?

Weyl’s Law

Exponent pairs

Evidence

Higher dimensions

Open problems

Distribution of λ−3/4R(λ)

QuestionIs λ−3/4R(λ) in B2 Besicovitch class?

Theorem (Khosravi 2006)∫ T

0R(x)3 dx ∼ cT 13/4, T →∞, c > 0

Heath-Brown (1992)

∃f (a) for Gauss and g(a) for Dirichlet divisor:

Xmeasure{x ∈ [1,X ], x−1/4R(x) ∈ I} →

∫If (a) da

Xmeasure{x ∈ [1,X ], x−1/4∆(x) ∈ I} →

∫Ig(a) da

weyl’s law for heisenberg manifolds chung, khosravi...

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