### 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_{t}f−
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_{t}f(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_{t}f−
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_{t}f(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_{t}f−
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_{t}f(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.

^{p}, 1≤p<∞

t→∞lim

λ_{t}f−
Z

X

fdµ p

=0.

^{p},p>1, and for almost everyx ∈X,

t→∞lim λ_{t}f(x) =
Z

X

fdµ .

_{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_{t}f −
Z

X

fdµ
_{p}

≤C_{p}m(B_{t})^{−θ}^{p}kfk_{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_{t}f(x)−
Z

X

fdµ

≤Cp(x,f)m(Bt)^{−θ}^{p}.
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_{t}f −
Z

X

fdµ
_{p}

≤C_{p}m(B_{t})^{−θ}^{p}kfk_{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_{t}f(x)−
Z

X

fdµ

≤Cp(x,f)m(Bt)^{−θ}^{p}.
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_{t}f −
Z

X

fdµ
_{p}

≤C_{p}m(B_{t})^{−θ}^{p}kfk_{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_{t}f(x)−
Z

X

fdµ

≤Cp(x,f)m(Bt)^{−θ}^{p}.
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_{t}f −
Z

X

fdµ 2

≤Cm(Bt)^{−θ}^{2}kfk_{2} ,

### 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_{t}f −
Z

X

fdµ 2

≤Cm(Bt)^{−θ}^{2}kfk_{2} ,

### 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_{t}f −
Z

X

fdµ 2

≤Cm(Bt)^{−θ}^{2}kfk_{2} ,

### 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 sphereS^{2}via 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 ofS^{2}which 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 sphereS^{2}via 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 ofS^{2}which 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 sphereS^{2}via 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 ofS^{2}which 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 =_{|B}^{1}

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_{t}onG.

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 =_{|B}^{1}

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_{t}onG.

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 =_{|B}^{1}

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_{t}onG.

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 =_{|B}^{1}

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.

_{t}onG.

### Averaging operators in general unitary representations

LetGbe an lcsc group,B_{t}⊂Gwithm_{G}(B_{t})→ ∞, andβ_{t} the
uniform measure onB_{t}.

_{X}(β_{t}) :H → H, given by : π(β_{t})v =_{|B}^{1}

t|

R

_{G}(g)
Our goal is to outline the proof of the ergodic theorems for lattice
subgroups of simple algebraic groupsG.

_{t}onG.

### Averaging operators in general unitary representations

LetGbe an lcsc group,B_{t}⊂Gwithm_{G}(B_{t})→ ∞, andβ_{t} the
uniform measure onB_{t}.

_{X}(β_{t}) :H → H, given by : π(β_{t})v =_{|B}^{1}

t|

R

_{G}(g)
Our goal is to outline the proof of the ergodic theorems for lattice
subgroups of simple algebraic groupsG.

_{t}onG.

### 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_{B}_{t}/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})^{−}^{2ne}^{1}^{(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_{B}_{t}/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})^{−}^{2ne}^{1}^{(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 onL^{2}_{0}(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
L^{2}(X)

≤Cθ(m_{G}(B_{t}))^{−θ}kfk_{2} ,

for every 0< θ < ^{1}

2ne(π_{X}^{0}).

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 onL^{2}_{0}(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
L^{2}(X)

≤Cθ(m_{G}(B_{t}))^{−θ}kfk_{2} ,

for every 0< θ < ^{1}

2ne(π_{X}^{0}).

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 onL^{2}_{0}(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
L^{2}(X)

≤Cθ(m_{G}(B_{t}))^{−θ}kfk_{2} ,

for every 0< θ < ^{1}

2ne(π_{X}^{0}).

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

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.

|γΛ∩BT| m(BT) = 1

[Γ : Λ] +O m(BT)^{−δ}

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,t_{0}andε_{0}such that for
allt≥t_{0}andε < ε_{0}we 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,t_{0}andε_{0}such that for
allt≥t_{0}andε < ε_{0}we 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,t_{0}andε_{0}such that for
allt≥t_{0}andε < ε_{0}we 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} inL^{2}(m_{G/Γ}):

π(β_{t})f −
Z

X

f dm_{G/Γ}
L^{2}

→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_{t}in
L^{2}(m_{G/Γ})satisfies

π(β_{t})f −
Z

G/Γ

f dm_{G/Γ}
L^{2}

≤Cm(B_{t})^{−θ}kfk_{L}2

Then |Γ_{t}|

m_{G}(B_{t}) =1+O

m(B_{t})^{−θ/(dim}^{G+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} inL^{2}(m_{G/Γ}):

π(β_{t})f −
Z

X

f dm_{G/Γ}
L^{2}

→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_{t}in
L^{2}(m_{G/Γ})satisfies

π(β_{t})f −
Z

G/Γ

f dm_{G/Γ}
L^{2}

≤Cm(B_{t})^{−θ}kfk_{L}2

Then |Γ_{t}|

m_{G}(B_{t}) =1+O

m(B_{t})^{−θ/(dim}^{G+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} inL^{2}(m_{G/Γ}):

π(β_{t})f−
Z

X

f dm_{G/Γ}
L^{2}

→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_{t}in
L^{2}(m_{G/Γ})satisfies

π(β_{t})f −
Z

G/Γ

f dm_{G/Γ}
L^{2}

≤Cm(B_{t})^{−θ}kfk_{L}2

Then |Γ_{t}|

m_{G}(B_{t}) =1+O

m(B_{t})^{−θ/(dim}^{G+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} inL^{2}(m_{G/Γ}):

π(β_{t})f−
Z

X

f dm_{G/Γ}
L^{2}

→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_{t}in
L^{2}(m_{G/Γ})satisfies

π(β_{t})f −
Z

G/Γ

f dm_{G/Γ}
L^{2}

≤Cm(B_{t})^{−θ}kfk_{L}2

Then |Γ_{t}|

m_{G}(B_{t}) =1+O

m(B_{t})^{−θ/(dim}^{G+1)}
.

Proof of the counting Theorem

Step 1 : Applying the mean ergodic theorem

LetOεbe a small neighborhood ofeand
χ_{ε}= χ_{O}_{ε}

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_{O}_{ε}

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_{O}_{ε}

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_{O}_{ε}

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^{−1}gtΓ)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_{ε}such
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^{−1}gtΓ)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_{ε}such
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^{−1}gtΓ)dm_{G}≤1+δ .

On the other hand, by definition ofφ_{ε}and the averaging operators
π_{G/Γ}(β_{t}):

π_{G/Γ}(β_{t})φε(hΓ) =

= 1

mG(Bt) Z

Bt

φε(g^{−1}hΓ)dm_{G}=

= 1

m_{G}(Bt)
Z

Bt

X

γ∈Γ

χε(g^{−1}hγ)dmG

=X

γ∈Γ

1 mG(Bt)

Z

Bt

χε(g^{−1}hγ)dm_{G}.

On the other hand, by definition ofφ_{ε}and the averaging operators
π_{G/Γ}(β_{t}):

π_{G/Γ}(β_{t})φε(hΓ) =

= 1

mG(Bt) Z

Bt

φε(g^{−1}hΓ)dm_{G}=

= 1

m_{G}(Bt)
Z

Bt

X

γ∈Γ

χε(g^{−1}hγ)dmG

=X

γ∈Γ

1 mG(Bt)

Z

Bt

χε(g^{−1}hγ)dm_{G}.

Step 2 : Basic comparison argument

Claim II.For sufficiently largetand anyh∈ O_{ε},
Z

Bt−cε

φε(g^{−1}hΓ)dm_{G}(g)≤ |Γ_{t}| ≤
Z

Bt+cε

φε(g^{−1}hΓ)dm_{G}(g).

Proof. Ifχ_{ε}(g^{−1}hγ)6=0 for someg∈Bt−cεandh∈ O_{ε}, then
clearlyg^{−1}hγ∈supp χ_{ε}, and so

γ∈h^{−1}·Bt−cε·(supp χε)⊂B_{t}.
by admissibility.

Hence,

Z

Bt−cε

φ_{ε}(g^{−1}hΓ)dm_{G}(g)≤

≤X

γ∈Γt

Z

Bt

χε(g^{−1}hγ)dmG(g)≤ |Γ_{t}|.

Step 2 : Basic comparison argument

Claim II.For sufficiently largetand anyh∈ O_{ε},
Z

Bt−cε

φε(g^{−1}hΓ)dm_{G}(g)≤ |Γ_{t}| ≤
Z

Bt+cε

φε(g^{−1}hΓ)dm_{G}(g).

Proof. Ifχ_{ε}(g^{−1}hγ)6=0 for someg∈Bt−cεandh∈ O_{ε}, then
clearlyg^{−1}hγ∈supp χ_{ε}, and so

γ∈h^{−1}·Bt−cε·(supp χε)⊂B_{t}.
by admissibility.

Hence,

Z

Bt−cε

φ_{ε}(g^{−1}hΓ)dm_{G}(g)≤

≤X

γ∈Γt

Z

Bt

χε(g^{−1}hγ)dmG(g)≤ |Γ_{t}|.

Step 2 : Basic comparison argument

Claim II.For sufficiently largetand anyh∈ O_{ε},
Z

Bt−cε

φε(g^{−1}hΓ)dm_{G}(g)≤ |Γ_{t}| ≤
Z

Bt+cε

φε(g^{−1}hΓ)dm_{G}(g).

Proof. Ifχ_{ε}(g^{−1}hγ)6=0 for someg∈Bt−cεandh∈ Oε, then
clearlyg^{−1}hγ∈supp χ_{ε}, and so

γ∈h^{−1}·Bt−cε·(supp χε)⊂B_{t}.
by admissibility.

Hence,

Z

Bt−cε

φ_{ε}(g^{−1}hΓ)dm_{G}(g)≤

≤X

γ∈Γt

Z

Bt

χε(g^{−1}hγ)dmG(g)≤ |Γ_{t}|.

Step 2 : Basic comparison argument

Claim II.For sufficiently largetand anyh∈ O_{ε},
Z

Bt−cε

φε(g^{−1}hΓ)dm_{G}(g)≤ |Γ_{t}| ≤
Z

Bt+cε

φε(g^{−1}hΓ)dm_{G}(g).

Proof. Ifχ_{ε}(g^{−1}hγ)6=0 for someg∈Bt−cεandh∈ Oε, then
clearlyg^{−1}hγ∈supp χ_{ε}, and so

γ∈h^{−1}·Bt−cε·(supp χε)⊂B_{t}.
by admissibility.

Hence,

Z

Bt−cε

φ_{ε}(g^{−1}hΓ)dm_{G}(g)≤

≤X

γ∈Γt

Z

Bt

χε(g^{−1}hγ)dm_{G}(g)≤ |Γ_{t}|.

On the other hand, forγ∈Γ_{t} andh∈ O_{ε},

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

again by admissibility,

and sinceχ_{ε}≥0 andR

Gχ_{ε}dm=1 :
Z

Bt+cε

φε(g^{−1}hΓ)dm_{G}(g)≥

≥X

γ∈Γt

Z

Bt+cε

χ_{ε}(g^{−1}hγ)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^{−1}hγ)) =hγ(suppχε)^{−1}⊂Bt+cε

again by admissibility,
and sinceχ_{ε}≥0 andR

Gχ_{ε}dm=1 :
Z

Bt+cε

φε(g^{−1}hΓ)dm_{G}(g)≥

≥X

γ∈Γt

Z

Bt+cε

χ_{ε}(g^{−1}hγ)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^{−1}hγ)) =hγ(suppχε)^{−1}⊂Bt+cε

again by admissibility,
and sinceχ_{ε}≥0 andR

Gχ_{ε}dm=1 :
Z

Bt+cε

φε(g^{−1}hΓ)dm_{G}(g)≥

≥X

γ∈Γt

Z

Bt+cε

χ_{ε}(g^{−1}hγ)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
_{L}_{2}

≤Cm(Bt)^{−θ}kfk_{L}2

we must show

|Γ_{t}|

m_{G}(B_{t})=1+O

m(Bt)^{dim}^{−θ}^{G+1}
.

Proof. Clearly forεsmall,mG(Oε)∼ε^{n}.
and thus alsokχ_{ε}k^{2}_{2}∼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^{−2}ε^{−n}m(Bt)^{−2θ}.

Step 3 : Counting with an error term Assuming

π(β_{t})f −
Z

G/Γ

fdµ
_{L}_{2}

≤Cm(Bt)^{−θ}kfk_{L}2

we must show

|Γ_{t}|

m_{G}(B_{t}) =1+O

m(Bt)^{dim}^{−θ}^{G+1}
.

Proof. Clearly forεsmall,mG(Oε)∼ε^{n}.
and thus alsokχ_{ε}k^{2}_{2}∼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^{−2}ε^{−n}m(Bt)^{−2θ}.

Step 3 : Counting with an error term Assuming

π(β_{t})f −
Z

G/Γ

fdµ
_{L}_{2}

≤Cm(Bt)^{−θ}kfk_{L}2

we must show

|Γ_{t}|

m_{G}(B_{t}) =1+O

m(Bt)^{dim}^{−θ}^{G+1}
.

Proof. Clearly forεsmall,m_{G}(Oε)∼ε^{n}.
and thus alsokχ_{ε}k^{2}_{2}∼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^{−2}ε^{−n}m(Bt)^{−2θ}.

Step 3 : Counting with an error term Assuming

π(β_{t})f −
Z

G/Γ

fdµ
_{L}_{2}

≤Cm(Bt)^{−θ}kfk_{L}2

we must show

|Γ_{t}|

m_{G}(B_{t}) =1+O

m(Bt)^{dim}^{−θ}^{G+1}
.

Proof. Clearly forεsmall,m_{G}(Oε)∼ε^{n}.
and thus alsokχ_{ε}k^{2}_{2}∼ε^{−n}, wheren=dimG.

_{G/Γ}({x ∈G/Γ : |π_{G/Γ}(β_{t})φε(x)−1|> δ})

≤Cδ^{−2}ε^{−n}m(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)∼ε^{n}.

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)∼ε^{n}.

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)∼ε^{n}.

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.

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

^{n}.

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}),

_{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 spaceL^{2}(G/Λ).
This holds when the set of finite index subgroups satisfy property
τ, namely when the spectral gap appearing in the

representationsL^{2}_{0}(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 spaceL^{2}(G/Λ).

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

representationsL^{2}_{0}(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 spaceL^{2}(G/Λ).

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

representationsL^{2}_{0}(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

Indeed all that is needed is that the averaging operatorsπ(β_{t})
satisfy the same norm decay estimate in the spaceL^{2}(G/Λ).

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

representationsL^{2}_{0}(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).