weyl’s law for heisenberg manifolds chung, khosravi...
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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
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
The Heisenberg group
Heisenberg algebra:
hn = 〈X1, . . . ,Xn,Y1, . . . ,Yn,Z 〉
[Xi ,Xj ] = [Yi ,Yj ] = [Xi ,Z ] = [Yi ,Z ] = 0
[Xi ,Yj ] = δijZ
Heisenberg Group:
Hn =
1 x z
0 Inty
0 0 1
, x, y ∈ Rn, z ∈ R
X (x, y, z) =
0 x z0 0 ty0 0 0
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
Γr =
1 x z
0 Inty
0 0 1
, xi ∈ riZ, y ∈ Zn, z ∈ Z
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
]
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
The spectrum of Heisenberg manifolds
Deninger-Singhof, Colin de Verdiere, Gordon-WilsonLet Lr = {X (x, y, 0), xi ∈ riZ, y ∈ Zn}
J =
[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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
The spectrum of Heisenberg manifolds
Deninger-Singhof, Colin de Verdiere, Gordon-WilsonLet Lr = {X (x, y, 0), xi ∈ riZ, y ∈ Zn}
J =
[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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
The spectrum of Heisenberg manifolds
Deninger-Singhof, Colin de Verdiere, Gordon-WilsonLet Lr = {X (x, y, 0), xi ∈ riZ, y ∈ Zn}
J =
[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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
The spectrum of Heisenberg manifolds
Deninger-Singhof, Colin de Verdiere, Gordon-WilsonLet Lr = {X (x, y, 0), xi ∈ riZ, y ∈ Zn}
J =
[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.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]I 2π factors:
1
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)}
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]I 2π factors:
1
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)}
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]
I 2π factors:
1
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)}
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]I 2π factors:
1
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)}
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]I 2π factors:
1
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)}
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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 problems2 4 6 8 10
2
4
6
8
10
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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 problems2 4 6 8 10
2
4
6
8
10
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Gauß circle problem
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Gauß circle problem
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Conjecture (Montgomery)
∀ε, (ε, 1/2 + ε) is an exponent pair.
I Implies Lindelof H, Hardy’s conjecture
I Implies Conjecture 1 for Heisenberg
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Conjecture (Montgomery)
∀ε, (ε, 1/2 + ε) is an exponent pair.
I Implies Lindelof H, Hardy’s conjecture
I Implies Conjecture 1 for Heisenberg
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Conjecture (Montgomery)
∀ε, (ε, 1/2 + ε) is an exponent pair.
I Implies Lindelof H, Hardy’s conjecture
I Implies Conjecture 1 for Heisenberg
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Evidence
Theorem (Khosravi, Toth)
Cramer’s formula
limT→∞
1
T 5/2
∫ T
0R(λ)2 dλ = c > 0
Compare with Cramer (1922)
I For Gauss circle problem
limT→∞
1
T 3/2
∫ T
0R(λ)2 dλ =
1
3π2
∑n
(r(n)
n3/4
)2
I For Dirichlet divisor problem
limT→∞
1
T 3/2
∫ T
0∆(λ)2 dλ =
1
6π2
∑n
(d(n)
n3/4
)2
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Evidence
Theorem (Khosravi, Toth)
Cramer’s formula
limT→∞
1
T 5/2
∫ T
0R(λ)2 dλ = c > 0
Compare with Cramer (1922)
I For Gauss circle problem
limT→∞
1
T 3/2
∫ T
0R(λ)2 dλ =
1
3π2
∑n
(r(n)
n3/4
)2
I For Dirichlet divisor problem
limT→∞
1
T 3/2
∫ T
0∆(λ)2 dλ =
1
6π2
∑n
(d(n)
n3/4
)2
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Evidence
Theorem (Khosravi, Toth)
Cramer’s formula
limT→∞
1
T 5/2
∫ T
0R(λ)2 dλ = c > 0
Compare with Cramer (1922)
I For Gauss circle problem
limT→∞
1
T 3/2
∫ T
0R(λ)2 dλ =
1
3π2
∑n
(r(n)
n3/4
)2
I For Dirichlet divisor problem
limT→∞
1
T 3/2
∫ T
0∆(λ)2 dλ =
1
6π2
∑n
(d(n)
n3/4
)2
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Evidence
Theorem (Khosravi, Toth)
Cramer’s formula
limT→∞
1
T 5/2
∫ T
0R(λ)2 dλ = c > 0
Compare with Cramer (1922)
I For Gauss circle problem
limT→∞
1
T 3/2
∫ T
0R(λ)2 dλ =
1
3π2
∑n
(r(n)
n3/4
)2
I For Dirichlet divisor problem
limT→∞
1
T 3/2
∫ T
0∆(λ)2 dλ =
1
6π2
∑n
(d(n)
n3/4
)2
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
∫ 1
0
∫ 1
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
∫ 1
0
∫ 1
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
∫ 1
0
∫ 1
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)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Lower bounds
Theorem (P., Toth)
For fixed g1
T
∫ 2T
T|R(λ)| dλ� T 3/4
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
E(t)
Error Term for N(t)
Figure: The relative error for the standard lattice E (t)/t3/4
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
−8000
−6000
−4000
−2000
0
2000
4000
6000
8000
10000
t
E(t)
Error Term for N(t)
Figure: The absolute error for the standard lattice
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Error Term for NL(t)
E(t)/
t3/4
t
Figure: The relative error for the lattice L
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
−4
−3
−2
−1
0
1
2
3
4
5
6
x
Delta
(x) /
x1/
4
Error term for the Dirichlet Divisor Problem
Figure: The relative error in the Dirichlet divisor problem
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
−80
−60
−40
−20
0
20
40
60
80
100
120
x
Delta
(x)
Error term for the Dirichlet Divisor Problem
Figure: The absolute error in the Dirichlet divisor problem
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
−8
−6
−4
−2
0
2
4
6Error Term for Gauss Circle Problem
E(x)
/x1/
4
x
Figure: The relative error term in Gauss’ circle problem
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
1000
2000
3000
4000
5000
6000
7000
8000Histogram for N(t) (normalized)
Figure: The Histogram for the standard lattice
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
1000
2000
3000
4000
5000
6000
7000
8000
Histogram for NL(t) (normalized)
Figure: Histogram for the Lattice L
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
−4 −3 −2 −1 0 1 2 3 4 5 60
1000
2000
3000
4000
5000
6000
7000
8000Histogram for Dirichlet Divisor Problem (normalized)
Figure: Histogram for Dirichlet’s Divisor problem
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
−8 −6 −4 −2 0 2 4 60
1000
2000
3000
4000
5000
6000
7000
8000Histogram for Gauss Circle Problem (normalized)
Figure: Histogram for Gauss’ circle problem
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Higher dimensions
Hormander: R(λ) = O(λn) on Hn
If d2i /d
2j are all rational, we call g rational.
Theorem (Khosravi, P.)
n > 1
I Rational case: R(λ) = O(λn−7/41)
I Irrational case: R(λ) = O(λn−1/4+ε) for generic g
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Higher dimensions
Hormander: R(λ) = O(λn) on Hn
If d2i /d
2j are all rational, we call g rational.
Theorem (Khosravi, P.)
n > 1
I Rational case: R(λ) = O(λn−7/41)
I Irrational case: R(λ) = O(λn−1/4+ε) for generic g
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Higher dimensions
Hormander: R(λ) = O(λn) on Hn
If d2i /d
2j are all rational, we call g rational.
Theorem (Khosravi, P.)
n > 1
I Rational case: R(λ) = O(λn−7/41)
I Irrational case: R(λ) = O(λn−1/4+ε) for generic g
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
Higher dimensions
Hormander: R(λ) = O(λn) on Hn
If d2i /d
2j are all rational, we call g rational.
Theorem (Khosravi, P.)
n > 1
I Rational case: R(λ) = O(λn−7/41)
I Irrational case: R(λ) = O(λn−1/4+ε) for generic g
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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
x>√
λ
y =∑
y≤√
λ
y∑
√λ<x≤λ/y
1 =∑
y≤√
λ
y([λ/y ]− [√λ])
=∑
y≤√
λ
y(λ/y −√λ− ψ(λ/y) + ψ(
√λ))
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
where ψ(u) = u − [u]− 1/2
y
x
1
4
0.5
02
-0.5
-1
0-2-4
ψ(u) = −∑n 6=0
e2πinu
2πin
Use Euler summation∑n≤X
n =X 2
2− ψ(X )X + O(1)
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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:
1
Xmeasure{x ∈ [1,X ], x−1/4R(x) ∈ I} →
∫If (a) da
1
Xmeasure{x ∈ [1,X ], x−1/4∆(x) ∈ I} →
∫Ig(a) da
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
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
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:
1
Xmeasure{x ∈ [1,X ], x−1/4R(x) ∈ I} →
∫If (a) da
1
Xmeasure{x ∈ [1,X ], x−1/4∆(x) ∈ I} →
∫Ig(a) da