Representation theory, discrete lattice subgroups, effective ergodic theorems, and applications

Representation theory, discrete lattice subgroups,

effective ergodic theorems, and applications

May 30, 2016

Geometric Analysis on Discrete Groups RIMS workshop, Kyoto

Amos Nevo, Technion

Based on joint work with Alex Gorodnik, and on joint work with Anish Ghosh and Alex Gorodnik

Plan

Talk I: Averaging operators in dynamical systems, operator norm estimates, and effective ergodic theorems for lattice subgroups

Talk II: Unitary representations, the automorphic representation of a lattice subgroup, counting lattice points and applications Talk III: An effective form for the duality principle for

homogeneous spaces and some of its applications :

equidistribution of lattice orbits, and Diophantine approximation on algebraic varieties

Plan

Talk I: Averaging operators in dynamical systems, operator norm estimates, and effective ergodic theorems for lattice subgroups Talk II: Unitary representations, the automorphic representation of a lattice subgroup, counting lattice points and applications

Talk III: An effective form for the duality principle for homogeneous spaces and some of its applications :

equidistribution of lattice orbits, and Diophantine approximation on algebraic varieties

Plan

Talk I: Averaging operators in dynamical systems, operator norm estimates, and effective ergodic theorems for lattice subgroups Talk II: Unitary representations, the automorphic representation of a lattice subgroup, counting lattice points and applications Talk III: An effective form for the duality principle for

homogeneous spaces and some of its applications :

equidistribution of lattice orbits, and Diophantine approximation on algebraic varieties

Ergodic theorems for general lattice actions

Ergodic theorems for lattice subgroups, I.Gorodnik+N, ’08.

For anarbitraryergodicΓ-action on a probability space(X, µ), the mean ergodic theorem holds: for everyf ∈L^p, 1≤p<∞

t→∞lim

λ_tf− Z

X

fdµ p

=0.

Furthermore, the pointwise ergodic theorem holds, namely for everyf ∈L^p,p>1, and for almost everyx ∈X,

t→∞lim λ_tf(x) = Z

X

fdµ .

We emphasize that this result holds forallΓ-actions. The only connection to the original embedding ofΓin the groupGis in the definition of the setsΓ_t.

Ergodic theorems for general lattice actions

Ergodic theorems for lattice subgroups, I.Gorodnik+N, ’08.

For anarbitraryergodicΓ-action on a probability space(X, µ), the mean ergodic theorem holds: for everyf ∈L^p, 1≤p<∞

t→∞lim

λ_tf− Z

X

fdµ p

=0.

Furthermore, the pointwise ergodic theorem holds, namely for everyf ∈L^p,p>1, and for almost everyx ∈X,

t→∞lim λ_tf(x) = Z

X

fdµ .

We emphasize that this result holds forallΓ-actions. The only connection to the original embedding ofΓin the groupGis in the definition of the setsΓ_t.

Ergodic theorems for general lattice actions

Ergodic theorems for lattice subgroups, I.Gorodnik+N, ’08.

For anarbitraryergodicΓ-action on a probability space(X, µ), the mean ergodic theorem holds: for everyf ∈L^p, 1≤p<∞

t→∞lim

λ_tf− Z

X

fdµ p

=0.

Furthermore, the pointwise ergodic theorem holds, namely for everyf ∈L^p,p>1, and for almost everyx ∈X,

t→∞lim λ_tf(x) = Z

X

fdµ .

We emphasize that this result holds forallΓ-actions. The only connection to the original embedding ofΓin the groupGis in the definition of the setsΓ_t.

Ergodic theorems for general lattice actions

Ergodic theorems for lattice subgroups, I.Gorodnik+N, ’08.

For anarbitraryergodicΓ-action on a probability space(X, µ), the mean ergodic theorem holds: for everyf ∈L^p, 1≤p<∞

t→∞lim

λ_tf− Z

X

fdµ p

=0.

Furthermore, the pointwise ergodic theorem holds, namely for everyf ∈L^p,p>1, and for almost everyx ∈X,

t→∞lim λ_tf(x) = Z

X

fdµ .

We emphasize that this result holds forallΓ-actions. The only connection to the original embedding ofΓin the groupGis in the definition of the setsΓ_t.

Spectral gap and the ultimate ergodic theorem

Ergodic theorems for lattice subgroups, II.Gorodnik+N, ’08.

If theΓ-action has a spectral gap then, the effective mean ergodic theorem holds : for everyf ∈L^p, 1<p<∞

λ_tf − Z

X

fdµ _p

≤C_pm(B_t)^−θpkfk_p,

whereθ_p=θ_p(X)>0.

Under this condition, the effective pointwise ergodic theorem holds: for everyf ∈L^p,p>1, for almost everyx,

λ_tf(x)− Z

X

fdµ

≤Cp(x,f)m(Bt)^−θp. whereθ_p=θ_p(X)>0.

Spectral gap and the ultimate ergodic theorem

Ergodic theorems for lattice subgroups, II.Gorodnik+N, ’08.

If theΓ-action has a spectral gap then, the effective mean ergodic theorem holds : for everyf ∈L^p, 1<p<∞

λ_tf − Z

X

fdµ _p

≤C_pm(B_t)^−θpkfk_p,

whereθ_p=θ_p(X)>0.

Under this condition, the effective pointwise ergodic theorem holds: for everyf ∈L^p,p>1, for almost everyx,

λ_tf(x)− Z

X

fdµ

≤Cp(x,f)m(Bt)^−θp. whereθ_p=θ_p(X)>0.

Spectral gap and the ultimate ergodic theorem

Ergodic theorems for lattice subgroups, II.Gorodnik+N, ’08.

If theΓ-action has a spectral gap then, the effective mean ergodic theorem holds : for everyf ∈L^p, 1<p<∞

λ_tf − Z

X

fdµ _p

≤C_pm(B_t)^−θpkfk_p,

whereθ_p=θ_p(X)>0.

Under this condition, the effective pointwise ergodic theorem holds: for everyf ∈L^p,p>1, for almost everyx,

λ_tf(x)− Z

X

fdµ

≤Cp(x,f)m(Bt)^−θp. whereθ_p=θ_p(X)>0.

Examples

In particular, ifΓhas propertyT, then the quantitative mean and pointwise ergodic theorems hold inevery ergodic

measure-preserving actionwith a fixedθ_p=θ_p(G)>0 independent ofX.

Letσ: Γ→U_n(C)be aunitary representation with dense image. Then for every unit vectoruthe setsσ(Γ_t)ubecome

equidistributed in the unit sphere, w.r.t. the rotation invariant measure, and the rate of equidistribution at every point is exponentially fast if the representations admits a spectral gap and the function is Holder.

Specializing further, in every action ofΓon afinite homogeneous spaceX, we have the following norm bound for the averaging operators

λ_tf − Z

X

fdµ 2

≤Cm(Bt)^−θ2kfk₂ ,

Examples

In particular, ifΓhas propertyT, then the quantitative mean and pointwise ergodic theorems hold inevery ergodic

measure-preserving actionwith a fixedθ_p=θ_p(G)>0 independent ofX.

Letσ: Γ→U_n(C)be aunitary representation with dense image.

Then for every unit vectoruthe setsσ(Γ_t)ubecome

equidistributed in the unit sphere, w.r.t. the rotation invariant measure, and the rate of equidistribution at every point is exponentially fast if the representations admits a spectral gap and the function is Holder.

Specializing further, in every action ofΓon afinite homogeneous spaceX, we have the following norm bound for the averaging operators

λ_tf − Z

X

fdµ 2

≤Cm(Bt)^−θ2kfk₂ ,

Examples

In particular, ifΓhas propertyT, then the quantitative mean and pointwise ergodic theorems hold inevery ergodic

measure-preserving actionwith a fixedθ_p=θ_p(G)>0 independent ofX.

Letσ: Γ→U_n(C)be aunitary representation with dense image.

Then for every unit vectoruthe setsσ(Γ_t)ubecome

equidistributed in the unit sphere, w.r.t. the rotation invariant measure, and the rate of equidistribution at every point is exponentially fast if the representations admits a spectral gap and the function is Holder.

Specializing further, in every action ofΓon afinite homogeneous spaceX, we have the following norm bound for the averaging operators

λ_tf − Z

X

fdµ 2

≤Cm(Bt)^−θ2kfk₂ ,

Discrete groups : some steps in the spectral approach

The problem of ergodic theorems for general discrete groups was raised already by Arnol’d and Krylov (1962). They proved an equidistribution theorem for dense free subgroups of isometries of the unit sphereS²via a spectral argument similar to Weyl’s equidistribution theorem on the circle (1918).

Guivarc’h has established amean ergodic theorem for radial averages on the free group, using von-Neumann’s original approach via the spectral theorem (1968).

The distinctly non-amenable possibility of ergodic theorems with quantitative estimates on the rate of convergence was realized first by the Lubotzky-Phillips-Sarnak construction of adense free group os isometries ofS²which has an optimal (!) spectral gap (1980’s).

Discrete groups : some steps in the spectral approach

The problem of ergodic theorems for general discrete groups was raised already by Arnol’d and Krylov (1962). They proved an equidistribution theorem for dense free subgroups of isometries of the unit sphereS²via a spectral argument similar to Weyl’s equidistribution theorem on the circle (1918).

Guivarc’h has established amean ergodic theorem for radial averages on the free group, using von-Neumann’s original approach via the spectral theorem (1968).

The distinctly non-amenable possibility of ergodic theorems with quantitative estimates on the rate of convergence was realized first by the Lubotzky-Phillips-Sarnak construction of adense free group os isometries ofS²which has an optimal (!) spectral gap (1980’s).

Discrete groups : some steps in the spectral approach

The problem of ergodic theorems for general discrete groups was raised already by Arnol’d and Krylov (1962). They proved an equidistribution theorem for dense free subgroups of isometries of the unit sphereS²via a spectral argument similar to Weyl’s equidistribution theorem on the circle (1918).

Guivarc’h has established amean ergodic theorem for radial averages on the free group, using von-Neumann’s original approach via the spectral theorem (1968).

The distinctly non-amenable possibility of ergodic theorems with quantitative estimates on the rate of convergence was realized first by the Lubotzky-Phillips-Sarnak construction of adense free group os isometries ofS²which has an optimal (!) spectral gap (1980’s).

Averaging operators in general unitary representations

LetGbe an lcsc group,B_t⊂Gwithm_G(B_t)→ ∞, andβ_t the uniform measure onB_t.

Letπ:G→ U(Hπ)be a strongly continuous unitary representation ofGdefine theaveraging operators π_X(β_t) :H → H, given by : π(β_t)v =_|B¹

t|

R

Btπ(g)v dm_G(g) Our goal is to outline the proof of the ergodic theorems for lattice subgroups of simple algebraic groupsG.

A key role is played by a fundamental norm estimate (and resulting mean ergodic theorem) which is satisfied by general families of averagesβ_tonG.

This estimate, which we now state, utilizes the unitary representation theory of simple algebraic groups.

Averaging operators in general unitary representations

LetGbe an lcsc group,B_t⊂Gwithm_G(B_t)→ ∞, andβ_t the uniform measure onB_t.

Letπ:G→ U(Hπ)be a strongly continuous unitary representation ofGdefine theaveraging operators π_X(β_t) :H → H, given by :

π(β_t)v =_|B¹

t|

R

Btπ(g)v dm_G(g) Our goal is to outline the proof of the ergodic theorems for lattice subgroups of simple algebraic groupsG.

A key role is played by a fundamental norm estimate (and resulting mean ergodic theorem) which is satisfied by general families of averagesβ_tonG.

This estimate, which we now state, utilizes the unitary representation theory of simple algebraic groups.

Averaging operators in general unitary representations

LetGbe an lcsc group,B_t⊂Gwithm_G(B_t)→ ∞, andβ_t the uniform measure onB_t.

Letπ:G→ U(Hπ)be a strongly continuous unitary representation ofGdefine theaveraging operators π_X(β_t) :H → H, given by : π(β_t)v =_|B¹

t|

R

Btπ(g)v dm_G(g)

Our goal is to outline the proof of the ergodic theorems for lattice subgroups of simple algebraic groupsG.

A key role is played by a fundamental norm estimate (and resulting mean ergodic theorem) which is satisfied by general families of averagesβ_tonG.

This estimate, which we now state, utilizes the unitary representation theory of simple algebraic groups.

Averaging operators in general unitary representations

LetGbe an lcsc group,B_t⊂Gwithm_G(B_t)→ ∞, andβ_t the uniform measure onB_t.

Letπ:G→ U(Hπ)be a strongly continuous unitary representation ofGdefine theaveraging operators π_X(β_t) :H → H, given by : π(β_t)v =_|B¹

t|

R

Btπ(g)v dm_G(g) Our goal is to outline the proof of the ergodic theorems for lattice subgroups of simple algebraic groupsG.

A key role is played by a fundamental norm estimate (and resulting mean ergodic theorem) which is satisfied by general families of averagesβ_tonG.

This estimate, which we now state, utilizes the unitary representation theory of simple algebraic groups.

Averaging operators in general unitary representations

LetGbe an lcsc group,B_t⊂Gwithm_G(B_t)→ ∞, andβ_t the uniform measure onB_t.

Letπ:G→ U(Hπ)be a strongly continuous unitary representation ofGdefine theaveraging operators π_X(β_t) :H → H, given by : π(β_t)v =_|B¹

t|

R

Btπ(g)v dm_G(g) Our goal is to outline the proof of the ergodic theorems for lattice subgroups of simple algebraic groupsG.

A key role is played by a fundamental norm estimate (and resulting mean ergodic theorem) which is satisfied by general families of averagesβ_tonG.

This estimate, which we now state, utilizes the unitary representation theory of simple algebraic groups.

Averaging operators in general unitary representations

LetGbe an lcsc group,B_t⊂Gwithm_G(B_t)→ ∞, andβ_t the uniform measure onB_t.

Letπ:G→ U(Hπ)be a strongly continuous unitary representation ofGdefine theaveraging operators π_X(β_t) :H → H, given by : π(β_t)v =_|B¹

t|

R

Btπ(g)v dm_G(g) Our goal is to outline the proof of the ergodic theorems for lattice subgroups of simple algebraic groupsG.

A key role is played by a fundamental norm estimate (and resulting mean ergodic theorem) which is satisfied by general families of averagesβ_tonG.

This estimate, which we now state, utilizes the unitary representation theory of simple algebraic groups.

Spectral transfer principle

Spectral transfer principle.[N 98], [Gorodnik+N 2005]. Forevery unitary representationπof a simple algebraic groupGwith a spectral gap and no finite-dimensional invariant subspaces, and forevery family of probability measuresβ_t=χ_Bt/mG(Bt), the following norm decay estimate holds (for every >0)

kπ(β_t)k ≤ kr_G(β_t)k^ne(π)1 ≤Cεm(B_t)^{−2ne1(π)+ε}, withne(π)a positive integer depending onπ.

The norm estimate of the operatorπ(β)in ageneral rep’π, has been reduced to a norm estimate for the convolution operator r_G(β)in theregular rep’r_G. This establishes for simple groups an analog of the transfer(ence) principle for amenable groups.

Spectral transfer principle

Spectral transfer principle.[N 98], [Gorodnik+N 2005]. Forevery unitary representationπof a simple algebraic groupGwith a spectral gap and no finite-dimensional invariant subspaces, and forevery family of probability measuresβ_t=χ_Bt/mG(Bt), the following norm decay estimate holds (for every >0)

kπ(β_t)k ≤ kr_G(β_t)k^ne(π)1 ≤Cεm(B_t)^{−2ne1(π)+ε}, withne(π)a positive integer depending onπ.

The norm estimate of the operatorπ(β)in ageneral rep’π, has been reduced to a norm estimate for the convolution operator r_G(β)in theregular rep’r_G. This establishes for simple groups an analog of the transfer(ence) principle for amenable groups.

The effective mean ergodic theorem

In particular, we can bound the norm of the averaging operators ofπ_X(β_t)acting onL²₀(X), when the action is ergodic and weak mixing, namely has no finite-dimensional invariant subspaces.

Thm. D.Effective mean ergodic theorem.[N 98], [Gorodnik+N 2005]. For any weak mixing action of a simple algebraic groupG which has a spectral gap,and for any familyBt ⊂Gwith

m_G(Bt)→ ∞, the convergence of the time averagesπ_X(β_t)to the space average takes place at a definite rate :

π(β_t)f − Z

X

fdµ L²(X)

≤Cθ(m_G(B_t))^−θkfk₂ ,

for every 0< θ < ¹

2ne(π_X⁰).

Note that theonly requirementneeded to obtain the effective mean ergodic Thm’ is simply thatm(Bt)→ ∞, and the geometry of the sets is not relevant at all. This fact allows a great deal of flexibility in its application.

The effective mean ergodic theorem

In particular, we can bound the norm of the averaging operators ofπ_X(β_t)acting onL²₀(X), when the action is ergodic and weak mixing, namely has no finite-dimensional invariant subspaces.

Thm. D.Effective mean ergodic theorem.[N 98], [Gorodnik+N 2005]. For any weak mixing action of a simple algebraic groupG which has a spectral gap,and for any familyBt ⊂Gwith

m_G(Bt)→ ∞, the convergence of the time averagesπ_X(β_t)to the space average takes place at a definite rate :

π(β_t)f − Z

X

fdµ L²(X)

≤Cθ(m_G(B_t))^−θkfk₂ ,

for every 0< θ < ¹

2ne(π_X⁰).

Note that theonly requirementneeded to obtain the effective mean ergodic Thm’ is simply thatm(Bt)→ ∞, and the geometry of the sets is not relevant at all. This fact allows a great deal of flexibility in its application.

The effective mean ergodic theorem

In particular, we can bound the norm of the averaging operators ofπ_X(β_t)acting onL²₀(X), when the action is ergodic and weak mixing, namely has no finite-dimensional invariant subspaces.

Thm. D.Effective mean ergodic theorem.[N 98], [Gorodnik+N 2005]. For any weak mixing action of a simple algebraic groupG which has a spectral gap,and for any familyBt ⊂Gwith

m_G(Bt)→ ∞, the convergence of the time averagesπ_X(β_t)to the space average takes place at a definite rate :

π(β_t)f − Z

X

fdµ L²(X)

≤Cθ(m_G(B_t))^−θkfk₂ ,

for every 0< θ < ¹

2ne(π_X⁰).

Note that theonly requirementneeded to obtain the effective mean ergodic Thm’ is simply thatm(Bt)→ ∞, and the geometry of the sets is not relevant at all. This fact allows a great deal of flexibility in its application.

The lattice point counting problem

LetGbe an lcsc group andΓ⊂Gadiscrete lattice subgroup, namely a closed countable subgroup such thatG/Γhas a G-invariant probability measure.

The lattice point counting problemin domainsBt ⊂Gcalls for obtaining an asymptotic for the number of lattice points ofΓinBt, namely|Γ∩Bt|, ideally so that

1) Haar measurem(B_t)is themain termin the asymptotic, 2) There is anerror estimateof the form

|Γ∩B_t|

m(Bt) =1+O m(Bt)^−δ

whereδ >0 and is as large as possible, and is given in an explicit form,

The lattice point counting problem

LetGbe an lcsc group andΓ⊂Gadiscrete lattice subgroup, namely a closed countable subgroup such thatG/Γhas a G-invariant probability measure.

The lattice point counting problemin domainsB_t ⊂Gcalls for obtaining an asymptotic for the number of lattice points ofΓinBt, namely|Γ∩Bt|, ideally so that

1) Haar measurem(B_t)is themain termin the asymptotic, 2) There is anerror estimateof the form

|Γ∩B_t|

m(Bt) =1+O m(Bt)^−δ

whereδ >0 and is as large as possible, and is given in an explicit form,

The lattice point counting problem

LetGbe an lcsc group andΓ⊂Gadiscrete lattice subgroup, namely a closed countable subgroup such thatG/Γhas a G-invariant probability measure.

The lattice point counting problemin domainsB_t ⊂Gcalls for obtaining an asymptotic for the number of lattice points ofΓinBt, namely|Γ∩Bt|, ideally so that

1) Haar measurem(B_t)is themain termin the asymptotic,

2) There is anerror estimateof the form

|Γ∩B_t|

m(Bt) =1+O m(Bt)^−δ

whereδ >0 and is as large as possible, and is given in an explicit form,

The lattice point counting problem

LetGbe an lcsc group andΓ⊂Gadiscrete lattice subgroup, namely a closed countable subgroup such thatG/Γhas a G-invariant probability measure.

The lattice point counting problemin domainsB_t ⊂Gcalls for obtaining an asymptotic for the number of lattice points ofΓinBt, namely|Γ∩Bt|, ideally so that

1) Haar measurem(B_t)is themain termin the asymptotic, 2) There is anerror estimateof the form

|Γ∩B_t|

m(Bt) =1+O m(Bt)^−δ

whereδ >0 and is as large as possible, and is given in an explicit form,

3) the solution applies togeneral families of setsBt in a general family of groups, to allow for a wide variety of counting problems which arise in applications,

4) the solution should establish whether the error estimate can be taken asuniform over all (or some) finite index subgroups Λ⊂Γ, and over all their cosets, namely:

|γΛ∩B_T| m(BT) = 1

[Γ : Λ] +O m(BT)^−δ

withδand the implied constant independent of the finite index subgroupΛ, and the coset representativeγ.

First main point: A general solution obeying the 4 requirements above can be given for lattices in simple algebraic groups and general domainsBt, using a method based on the effective mean ergodic theorem forG.

As we shall see, this is the first essential step in proving mean ergodic theorems for lattice.

3) the solution applies togeneral families of setsBt in a general family of groups, to allow for a wide variety of counting problems which arise in applications,

4) the solution should establish whether the error estimate can be taken asuniform over all (or some) finite index subgroups Λ⊂Γ, and over all their cosets, namely:

|γΛ∩BT| m(BT) = 1

[Γ : Λ] +O m(BT)^−δ

withδand the implied constant independent of the finite index subgroupΛ, and the coset representativeγ.

First main point: A general solution obeying the 4 requirements above can be given for lattices in simple algebraic groups and general domainsBt, using a method based on the effective mean ergodic theorem forG.

As we shall see, this is the first essential step in proving mean ergodic theorems for lattice.

3) the solution applies togeneral families of setsBt in a general family of groups, to allow for a wide variety of counting problems which arise in applications,

4) the solution should establish whether the error estimate can be taken asuniform over all (or some) finite index subgroups Λ⊂Γ, and over all their cosets, namely:

|γΛ∩BT| m(BT) = 1

[Γ : Λ] +O m(BT)^−δ

withδand the implied constant independent of the finite index subgroupΛ, and the coset representativeγ.

First main point: A general solution obeying the 4 requirements above can be given for lattices in simple algebraic groups and general domainsBt, using a method based on the effective mean ergodic theorem forG.

As we shall see, this is the first essential step in proving mean ergodic theorems for lattice.

3) the solution applies togeneral families of setsBt in a general family of groups, to allow for a wide variety of counting problems which arise in applications,

4) the solution should establish whether the error estimate can be taken asuniform over all (or some) finite index subgroups Λ⊂Γ, and over all their cosets, namely:

|γΛ∩BT| m(BT) = 1

[Γ : Λ] +O m(BT)^−δ

withδand the implied constant independent of the finite index subgroupΛ, and the coset representativeγ.

First main point: A general solution obeying the 4 requirements above can be given for lattices in simple algebraic groups and general domainsBt, using a method based on the effective mean ergodic theorem forG.

As we shall see, this is the first essential step in proving mean ergodic theorems for lattice.

Admissible sets for the counting problem

Somestability and regularity assumptionson the setsBtare necessary for the lattice point counting problem.

AssumeGis a simple Lie group, fix any left-invariant Riemannian metric onG, and let

O_ε={g∈G: d(g,e)< ε}.

An increasing family of bounded Borel subsetB_t,t >0, ofGwill be calledadmissibleif there existsc>0,t₀andε₀such that for allt≥t₀andε < ε₀we have :

O_ε·Bt· O_ε⊂Bt+cε, (1) m_G(B_t+ε)≤(1+cε)·m_G(B_t). (2)

Admissible sets for the counting problem

Somestability and regularity assumptionson the setsBtare necessary for the lattice point counting problem.

AssumeGis a simple Lie group, fix any left-invariant Riemannian metric onG, and let

O_ε={g∈G: d(g,e)< ε}.

An increasing family of bounded Borel subsetB_t,t >0, ofGwill be calledadmissibleif there existsc>0,t₀andε₀such that for allt≥t₀andε < ε₀we have :

O_ε·Bt· O_ε⊂Bt+cε, (1) m_G(B_t+ε)≤(1+cε)·m_G(B_t). (2)

Admissible sets for the counting problem

Somestability and regularity assumptionson the setsBtare necessary for the lattice point counting problem.

AssumeGis a simple Lie group, fix any left-invariant Riemannian metric onG, and let

O_ε={g∈G: d(g,e)< ε}.

An increasing family of bounded Borel subsetB_t,t>0, ofGwill be calledadmissibleif there existsc>0,t₀andε₀such that for allt≥t₀andε < ε₀we have :

O_ε·Bt· O_ε⊂Bt+cε, (1) m_G(B_t+ε)≤(1+cε)·m_G(B_t). (2)

Counting in connected Lie groups

Lattice point counting. Gorodnik+N, 2006.

Ga connected Lie group,Γ⊂Ga lattice,Bt⊂Gadmissible. 1) Assume the mean ergodic theorem holds forβ_t inL²(m_G/Γ):

π(β_t)f − Z

X

f dm_G/Γ L²

→0, (m_G/Γ(G/Γ) =1). Then

|Γ∩B_t|=|Γ_t| ∼m_G(B_t) as t → ∞.

Assume that the error term in the mean ergodic theorem forβ_tin L²(m_G/Γ)satisfies

π(β_t)f − Z

G/Γ

f dm_G/Γ L²

≤Cm(B_t)^−θkfk_L2

Then |Γ_t|

m_G(B_t) =1+O

m(B_t)^{−θ/(dimG+1)} .

Counting in connected Lie groups

Lattice point counting. Gorodnik+N, 2006.

Ga connected Lie group,Γ⊂Ga lattice,Bt⊂Gadmissible.

1) Assume the mean ergodic theorem holds forβ_t inL²(m_G/Γ):

π(β_t)f − Z

X

f dm_G/Γ L²

→0, (m_G/Γ(G/Γ) =1). Then

|Γ∩B_t|=|Γ_t| ∼m_G(B_t) as t → ∞.

Assume that the error term in the mean ergodic theorem forβ_tin L²(m_G/Γ)satisfies

π(β_t)f − Z

G/Γ

f dm_G/Γ L²

≤Cm(B_t)^−θkfk_L2

Then |Γ_t|

m_G(B_t) =1+O

m(B_t)^{−θ/(dimG+1)} .

Counting in connected Lie groups

Lattice point counting. Gorodnik+N, 2006.

Ga connected Lie group,Γ⊂Ga lattice,Bt⊂Gadmissible.

1) Assume the mean ergodic theorem holds forβ_t inL²(m_G/Γ):

π(β_t)f− Z

X

f dm_G/Γ L²

→0, (m_G/Γ(G/Γ) =1).

Then

|Γ∩B_t|=|Γ_t| ∼m_G(B_t) as t → ∞.

Assume that the error term in the mean ergodic theorem forβ_tin L²(m_G/Γ)satisfies

π(β_t)f − Z

G/Γ

f dm_G/Γ L²

≤Cm(B_t)^−θkfk_L2

Then |Γ_t|

m_G(B_t) =1+O

m(B_t)^{−θ/(dimG+1)} .

Counting in connected Lie groups

Lattice point counting. Gorodnik+N, 2006.

Ga connected Lie group,Γ⊂Ga lattice,Bt⊂Gadmissible.

1) Assume the mean ergodic theorem holds forβ_t inL²(m_G/Γ):

π(β_t)f− Z

X

f dm_G/Γ L²

→0, (m_G/Γ(G/Γ) =1).

Then

|Γ∩B_t|=|Γ_t| ∼m_G(B_t) as t → ∞.

Assume that the error term in the mean ergodic theorem forβ_tin L²(m_G/Γ)satisfies

π(β_t)f − Z

G/Γ

f dm_G/Γ L²

≤Cm(B_t)^−θkfk_L2

Then |Γ_t|

m_G(B_t) =1+O

m(B_t)^{−θ/(dimG+1)} .

Proof of the counting Theorem

Step 1 : Applying the mean ergodic theorem

LetOεbe a small neighborhood ofeand χ_ε= χ_Oε

m_G(Oε)

consider theΓ-periodization ofχ_ε φ_ε(gΓ) =X

γ∈Γ

χ_ε(gγ).

Clearlyφis a bounded function onG/Γwith compact support, Z

G

χεdm_G=1, and Z

G/Γ

φεdµ_G/Γ=1.

Proof of the counting Theorem

Step 1 : Applying the mean ergodic theorem

LetOεbe a small neighborhood ofeand χ_ε= χ_Oε

m_G(Oε)

consider theΓ-periodization ofχ_ε φ_ε(gΓ) =X

γ∈Γ

χ_ε(gγ).

Clearlyφis a bounded function onG/Γwith compact support, Z

G

χεdm_G=1, and Z

G/Γ

φεdµ_G/Γ=1.

Proof of the counting Theorem

Step 1 : Applying the mean ergodic theorem

LetOεbe a small neighborhood ofeand χ_ε= χ_Oε

m_G(Oε)

consider theΓ-periodization ofχ_ε φ_ε(gΓ) =X

γ∈Γ

χ_ε(gγ).

Clearlyφis a bounded function onG/Γwith compact support, Z

G

χεdm_G=1, and Z

G/Γ

φεdµ_G/Γ=1.

Proof of the counting Theorem

Step 1 : Applying the mean ergodic theorem

LetOεbe a small neighborhood ofeand χ_ε= χ_Oε

m_G(Oε)

consider theΓ-periodization ofχ_ε φ_ε(gΓ) =X

γ∈Γ

χ_ε(gγ).

Clearlyφis a bounded function onG/Γwith compact support, Z

G

χεdm_G=1, and Z

G/Γ

φεdµ_G/Γ=1.

Let us apply the mean ergodic theorem to the functionφε. It follows from Chebycheff’s inequality that for everyδ >0, m_G/Γ({hΓ∈G/Γ : |π_G/Γ(β_t)φε(hΓ)−1|> δ})−→0

In particular, for sufficiently larget, the measure of the deviation set is smaller thanm_G/Γ(Oε), and so there existsgt ∈ Oεsuch that

|π_G/Γ(β_t)φε(gtΓ)−1| ≤δ and we can conclude the following

Claim I. Givenε, δ >0, fortsufficiently large, there exists g_t ∈ Oεsatisfying

1−δ≤ 1 m_G(B_t)

Z

Bt

φ_ε(g⁻¹gtΓ)dm_G≤1+δ .

Let us apply the mean ergodic theorem to the functionφε. It follows from Chebycheff’s inequality that for everyδ >0, m_G/Γ({hΓ∈G/Γ : |π_G/Γ(β_t)φε(hΓ)−1|> δ})−→0

In particular, for sufficiently larget, the measure of the deviation set is smaller thanm_G/Γ(O_ε), and so there existsgt ∈ O_εsuch that

|π_G/Γ(β_t)φε(gtΓ)−1| ≤δ and we can conclude the following

Claim I. Givenε, δ >0, fortsufficiently large, there exists g_t ∈ Oεsatisfying

1−δ≤ 1 m_G(B_t)

Z

Bt

φ_ε(g⁻¹gtΓ)dm_G≤1+δ .

Let us apply the mean ergodic theorem to the functionφε. It follows from Chebycheff’s inequality that for everyδ >0, m_G/Γ({hΓ∈G/Γ : |π_G/Γ(β_t)φε(hΓ)−1|> δ})−→0

In particular, for sufficiently larget, the measure of the deviation set is smaller thanm_G/Γ(O_ε), and so there existsgt ∈ O_εsuch that

|π_G/Γ(β_t)φε(gtΓ)−1| ≤δ and we can conclude the following

Claim I. Givenε, δ >0, fortsufficiently large, there exists g_t∈ Oεsatisfying

1−δ≤ 1 m_G(B_t)

Z

Bt

φ_ε(g⁻¹gtΓ)dm_G≤1+δ .

On the other hand, by definition ofφ_εand the averaging operators π_G/Γ(β_t):

π_G/Γ(β_t)φε(hΓ) =

= 1

mG(Bt) Z

Bt

φε(g⁻¹hΓ)dm_G=

= 1

m_G(Bt) Z

Bt

X

γ∈Γ

χε(g⁻¹hγ)dmG

=X

γ∈Γ

1 mG(Bt)

Z

Bt

χε(g⁻¹hγ)dm_G.

On the other hand, by definition ofφ_εand the averaging operators π_G/Γ(β_t):

π_G/Γ(β_t)φε(hΓ) =

= 1

mG(Bt) Z

Bt

φε(g⁻¹hΓ)dm_G=

= 1

m_G(Bt) Z

Bt

X

γ∈Γ

χε(g⁻¹hγ)dmG

=X

γ∈Γ

1 mG(Bt)

Z

Bt

χε(g⁻¹hγ)dm_G.

Step 2 : Basic comparison argument

Claim II.For sufficiently largetand anyh∈ O_ε, Z

Bt−cε

φε(g⁻¹hΓ)dm_G(g)≤ |Γ_t| ≤ Z

Bt+cε

φε(g⁻¹hΓ)dm_G(g).

Proof. Ifχ_ε(g⁻¹hγ)6=0 for someg∈Bt−cεandh∈ O_ε, then clearlyg⁻¹hγ∈supp χ_ε, and so

γ∈h⁻¹·Bt−cε·(supp χε)⊂B_t. by admissibility.

Hence,

Z

Bt−cε

φ_ε(g⁻¹hΓ)dm_G(g)≤

≤X

γ∈Γt

Z

Bt

χε(g⁻¹hγ)dmG(g)≤ |Γ_t|.

Step 2 : Basic comparison argument

Claim II.For sufficiently largetand anyh∈ O_ε, Z

Bt−cε

φε(g⁻¹hΓ)dm_G(g)≤ |Γ_t| ≤ Z

Bt+cε

φε(g⁻¹hΓ)dm_G(g).

Proof. Ifχ_ε(g⁻¹hγ)6=0 for someg∈Bt−cεandh∈ O_ε, then clearlyg⁻¹hγ∈supp χ_ε, and so

γ∈h⁻¹·Bt−cε·(supp χε)⊂B_t. by admissibility.

Hence,

Z

Bt−cε

φ_ε(g⁻¹hΓ)dm_G(g)≤

≤X

γ∈Γt

Z

Bt

χε(g⁻¹hγ)dmG(g)≤ |Γ_t|.

Step 2 : Basic comparison argument

Claim II.For sufficiently largetand anyh∈ O_ε, Z

Bt−cε

φε(g⁻¹hΓ)dm_G(g)≤ |Γ_t| ≤ Z

Bt+cε

φε(g⁻¹hΓ)dm_G(g).

Proof. Ifχ_ε(g⁻¹hγ)6=0 for someg∈Bt−cεandh∈ Oε, then clearlyg⁻¹hγ∈supp χ_ε, and so

γ∈h⁻¹·Bt−cε·(supp χε)⊂B_t. by admissibility.

Hence,

Z

Bt−cε

φ_ε(g⁻¹hΓ)dm_G(g)≤

≤X

γ∈Γt

Z

Bt

χε(g⁻¹hγ)dmG(g)≤ |Γ_t|.

Step 2 : Basic comparison argument

Claim II.For sufficiently largetand anyh∈ O_ε, Z

Bt−cε

φε(g⁻¹hΓ)dm_G(g)≤ |Γ_t| ≤ Z

Bt+cε

φε(g⁻¹hΓ)dm_G(g).

Proof. Ifχ_ε(g⁻¹hγ)6=0 for someg∈Bt−cεandh∈ Oε, then clearlyg⁻¹hγ∈supp χ_ε, and so

γ∈h⁻¹·Bt−cε·(supp χε)⊂B_t. by admissibility.

Hence,

Z

Bt−cε

φ_ε(g⁻¹hΓ)dm_G(g)≤

≤X

γ∈Γt

Z

Bt

χε(g⁻¹hγ)dm_G(g)≤ |Γ_t|.

On the other hand, forγ∈Γ_t andh∈ O_ε,

supp(g7→χε(g⁻¹hγ)) =hγ(suppχε)⁻¹⊂Bt+cε

again by admissibility,

and sinceχ_ε≥0 andR

Gχ_εdm=1 : Z

Bt+cε

φε(g⁻¹hΓ)dm_G(g)≥

≥X

γ∈Γt

Z

Bt+cε

χ_ε(g⁻¹hγ)dm_G(g)≥ |Γ_t|.

Now takingt sufficiently large,h=g_t and usingClaims I and II

|Γ_t| ≤(1+δ)m(B_t+ε)≤

≤(1+δ)(1+cε)m(B_t),

by admissibility. The lower estimate is proved similarly.

On the other hand, forγ∈Γ_t andh∈ O_ε,

supp(g7→χε(g⁻¹hγ)) =hγ(suppχε)⁻¹⊂Bt+cε

again by admissibility, and sinceχ_ε≥0 andR

Gχ_εdm=1 : Z

Bt+cε

φε(g⁻¹hΓ)dm_G(g)≥

≥X

γ∈Γt

Z

Bt+cε

χ_ε(g⁻¹hγ)dm_G(g)≥ |Γ_t|.

Now takingt sufficiently large,h=g_t and usingClaims I and II

|Γ_t| ≤(1+δ)m(B_t+ε)≤

≤(1+δ)(1+cε)m(B_t),

by admissibility. The lower estimate is proved similarly.

On the other hand, forγ∈Γ_t andh∈ O_ε,

supp(g7→χε(g⁻¹hγ)) =hγ(suppχε)⁻¹⊂Bt+cε

again by admissibility, and sinceχ_ε≥0 andR

Gχ_εdm=1 : Z

Bt+cε

φε(g⁻¹hΓ)dm_G(g)≥

≥X

γ∈Γt

Z

Bt+cε

χ_ε(g⁻¹hγ)dm_G(g)≥ |Γ_t|.

Now takingt sufficiently large,h=g_t and usingClaims I and II

|Γ_t| ≤(1+δ)m(B_t+ε)≤

≤(1+δ)(1+cε)m(B_t),

by admissibility. The lower estimate is proved similarly.

Step 3 : Counting with an error term

Assuming

π(β_t)f − Z

G/Γ

fdµ _L2

≤Cm(Bt)^−θkfk_L2

we must show

|Γ_t|

m_G(B_t)=1+O

m(Bt)^dim−θG+1 .

Proof. Clearly forεsmall,mG(Oε)∼εⁿ. and thus alsokχ_εk²₂∼ε⁻ⁿ, wheren=dimG.

By the mean ergodic theorem and Chebycheff’s inequality : m_G/Γ({x ∈G/Γ : |π_G/Γ(β_t)φε(x)−1|> δ})

≤Cδ⁻²ε⁻ⁿm(Bt)^−2θ.

Step 3 : Counting with an error term Assuming

π(β_t)f − Z

G/Γ

fdµ _L2

≤Cm(Bt)^−θkfk_L2

we must show

|Γ_t|

m_G(B_t) =1+O

m(Bt)^dim−θG+1 .

Proof. Clearly forεsmall,mG(Oε)∼εⁿ. and thus alsokχ_εk²₂∼ε⁻ⁿ, wheren=dimG.

By the mean ergodic theorem and Chebycheff’s inequality : m_G/Γ({x ∈G/Γ : |π_G/Γ(β_t)φε(x)−1|> δ})

≤Cδ⁻²ε⁻ⁿm(Bt)^−2θ.

Step 3 : Counting with an error term Assuming

π(β_t)f − Z

G/Γ

fdµ _L2

≤Cm(Bt)^−θkfk_L2

we must show

|Γ_t|

m_G(B_t) =1+O

m(Bt)^dim−θG+1 .

Proof. Clearly forεsmall,m_G(Oε)∼εⁿ. and thus alsokχ_εk²₂∼ε⁻ⁿ, wheren=dimG.

By the mean ergodic theorem and Chebycheff’s inequality : m_G/Γ({x ∈G/Γ : |π_G/Γ(β_t)φε(x)−1|> δ})

≤Cδ⁻²ε⁻ⁿm(Bt)^−2θ.

Step 3 : Counting with an error term Assuming

π(β_t)f − Z

G/Γ

fdµ _L2

≤Cm(Bt)^−θkfk_L2

we must show

|Γ_t|

m_G(B_t) =1+O

m(Bt)^dim−θG+1 .

Proof. Clearly forεsmall,m_G(Oε)∼εⁿ. and thus alsokχ_εk²₂∼ε⁻ⁿ, wheren=dimG.

By the mean ergodic theorem and Chebycheff’s inequality : m_G/Γ({x ∈G/Γ : |π_G/Γ(β_t)φε(x)−1|> δ})

≤Cδ⁻²ε⁻ⁿm(Bt)^−2θ.

Thus here the measure of the set of deviation decreases intwith a prescribed rate determined by the effectve ergodic Thm.

As we saw above, pointsx in its complement give us an

approximation to our counting problem with qualityδ, so we must require that the measure be smaller thanmG(Oε)∼εⁿ.

The estimate of the measure of the deviation set holds forall t,ε andδ, since the mean ergodic theorem with error term is a statement about the rate of convergence inoperator norm, and is thusuniformover all functions.

Our upper error estimate in the counting problem is, as before

|Γ_t| ≤(1+δ)m(B_t+ε)≤

≤(1+δ)(1+cε)m(B_t),

Takingδ∼ε∼m(B_t)^−θ/(n+1)to balance the two significant parts of the error appearing in the estimate(1+δ)(1+cε), the result follows.

Thus here the measure of the set of deviation decreases intwith a prescribed rate determined by the effectve ergodic Thm.

As we saw above, pointsx in its complement give us an

approximation to our counting problem with qualityδ, so we must require that the measure be smaller thanmG(Oε)∼εⁿ.

The estimate of the measure of the deviation set holds forall t,ε andδ, since the mean ergodic theorem with error term is a statement about the rate of convergence inoperator norm, and is thusuniformover all functions.

Our upper error estimate in the counting problem is, as before

|Γ_t| ≤(1+δ)m(B_t+ε)≤

≤(1+δ)(1+cε)m(B_t),

Takingδ∼ε∼m(B_t)^−θ/(n+1)to balance the two significant parts of the error appearing in the estimate(1+δ)(1+cε), the result follows.

Thus here the measure of the set of deviation decreases intwith a prescribed rate determined by the effectve ergodic Thm.

As we saw above, pointsx in its complement give us an

approximation to our counting problem with qualityδ, so we must require that the measure be smaller thanmG(Oε)∼εⁿ.

The estimate of the measure of the deviation set holds forall t,ε andδ, since the mean ergodic theorem with error term is a statement about the rate of convergence inoperator norm, and is thusuniformover all functions.

Our upper error estimate in the counting problem is, as before

|Γ_t| ≤(1+δ)m(B_t+ε)≤

≤(1+δ)(1+cε)m(B_t),

Takingδ∼ε∼m(B_t)^−θ/(n+1)to balance the two significant parts of the error appearing in the estimate(1+δ)(1+cε), the result follows.

Thus here the measure of the set of deviation decreases intwith a prescribed rate determined by the effectve ergodic Thm.

As we saw above, pointsx in its complement give us an

approximation to our counting problem with qualityδ, so we must require that the measure be smaller thanmG(Oε)∼εⁿ.

The estimate of the measure of the deviation set holds forall t,ε andδ, since the mean ergodic theorem with error term is a statement about the rate of convergence inoperator norm, and is thusuniformover all functions.

Our upper error estimate in the counting problem is, as before

|Γ_t| ≤(1+δ)m(B_t+ε)≤

≤(1+δ)(1+cε)m(B_t),

Takingδ∼ε∼m(B_t)^−θ/(n+1)to balance the two significant parts of the error appearing in the estimate(1+δ)(1+cε), the result follows.

Uniformity in the lattice point counting problem

Note that in the ergodic theoretic approach we have taken, the important feature of uniformity of counting lattice points in finite index subgroupsΛis apparent.

Indeed all that is needed is that the averaging operatorsπ(β_t) satisfy the same norm decay estimate in the spaceL²(G/Λ). This holds when the set of finite index subgroups satisfy property τ, namely when the spectral gap appearing in the

representationsL²₀(G/Λ)has a positive lower bound.

Propertyτhas been shown to hold for the set of congruence subgroups of any arithmetic lattice in a semisimple Lie group (Burger-Sarnak, Lubotzky, Clozel... generalizing the Selberg property)

.

Uniformity in the lattice point counting problem

Note that in the ergodic theoretic approach we have taken, the important feature of uniformity of counting lattice points in finite index subgroupsΛis apparent.

Indeed all that is needed is that the averaging operatorsπ(β_t) satisfy the same norm decay estimate in the spaceL²(G/Λ).

This holds when the set of finite index subgroups satisfy property τ, namely when the spectral gap appearing in the

representationsL²₀(G/Λ)has a positive lower bound.

Propertyτhas been shown to hold for the set of congruence subgroups of any arithmetic lattice in a semisimple Lie group (Burger-Sarnak, Lubotzky, Clozel... generalizing the Selberg property)

.

Uniformity in the lattice point counting problem

Note that in the ergodic theoretic approach we have taken, the important feature of uniformity of counting lattice points in finite index subgroupsΛis apparent.

Indeed all that is needed is that the averaging operatorsπ(β_t) satisfy the same norm decay estimate in the spaceL²(G/Λ).

This holds when the set of finite index subgroups satisfy property τ, namely when the spectral gap appearing in the

representationsL²₀(G/Λ)has a positive lower bound.

Propertyτhas been shown to hold for the set of congruence subgroups of any arithmetic lattice in a semisimple Lie group (Burger-Sarnak, Lubotzky, Clozel... generalizing the Selberg property)

.

Uniformity in the lattice point counting problem

Note that in the ergodic theoretic approach we have taken, the important feature of uniformity of counting lattice points in finite index subgroupsΛis apparent.

Indeed all that is needed is that the averaging operatorsπ(β_t) satisfy the same norm decay estimate in the spaceL²(G/Λ).

This holds when the set of finite index subgroups satisfy property τ, namely when the spectral gap appearing in the

representationsL²₀(G/Λ)has a positive lower bound.

Propertyτhas been shown to hold for the set of congruence subgroups of any arithmetic lattice in a semisimple Lie group (Burger-Sarnak, Lubotzky, Clozel... generalizing the Selberg property).