Annals of Mathematics
Pair correlation densities of
inhomogeneous quadratic forms By Jens Marklof

Annals of Mathematics, 158 (2003), 419–471
Pair correlation densities of
inhomogeneous quadratic forms
By Jens Marklof
Abstract
Under explicit diophantine conditions on (α, β) ∈
2
,weprove that the
local two-point correlations of the sequence given by the values (m − α)
2
+
(n−β)
2
, with (m, n) ∈
2
, are those of a Poisson process. This partly confirms
a conjecture of Berry and Tabor [2] on spectral statistics of quantized integrable
systems, and also establishes a particular case of the quantitative version of the
Oppenheim conjecture for inhomogeneous quadratic forms of signature (2,2).
The proof uses theta sums and Ratner’s classification of measures invariant
under unipotent flows.

2
∂x
2
−
∂
2
∂y
2
with quasi-periodicity conditions
ϕ(x + k,y + l)=e
−2πi(αk+βl)
ϕ(x, y),k,l∈ .
The corresponding classical dynamical system is the geodesic flow on the unit
tangent bundle of the flat torus
2
.
420 JENS MARKLOF
1.3. The asymptotic density of the sequence of λ
j
is π, according to the
well known formula for the number of lattice points in a large, shifted circle:
#{j : λ
j
≤ λ} =#{(m, n) ∈
2
:(m − α)
2
+(n − β)
2
≤ λ}∼πλ

The following result is classical.
1.5. Theorem. If the λ
j
come from a Poisson process with mean den-
sity D,
lim
λ→∞
R
2
[a, b](λ)=D(b −a)
almost surely.
1.6. We will assume throughout most of the paper that α, β, 1 are linearly
independent over
. This makes sure that there are no systematic degeneracies
in the sequence, which would contradict the independence we wish to estab-
lish. The symmetries leading to those degeneracies can, however, be removed
without much difficulty. This will be illustrated in Appendix A.
1.7. We shall need a mild diophantine condition on α.Anirrational
number α ∈
is called diophantine if there exist constants κ, C > 0 such that



α −
p
q



>

λ→∞
R
2
[a, b](λ)=π(b −a),
that is, on average over α, β.
1.11. Remark. Notice that Theorem 1.8 is much stronger than the corol-
lary. It provides explicit examples of “random” deterministic sequences that
satisfy the pair correlation conjecture. An admissible choice is for instance
α =
√
2, β =
√
3 [26].
1.12. The statement of Theorem 1.8 does not hold for any rational α, β,
where the pair correlation function is unbounded (see Appendix A.10 for de-
tails). This can be used to show that for generic (α, β) (in the topological
sense) the pair correlation function does not converge to a uniform density:
1.13. Theorem. For any a>0, there exists a set C ⊂
2
of second
Baire category, for which the following holds.
2
(i) For (α, β) ∈ C, there exist arbitrarily large λ such that
R
2
[−a, a](λ) ≥
log λ
log log log λ
.
(ii) For (α, β) ∈ C, there exists an infinite sequence L

They consider a slightly different statistic, the number of lattice points in a random circular
strip of fixed area. The variance of this distribution is very closely related to our pair correlation
function.
2
A set of first Baire category is a countable union of nowhere dense sets. Sets of second category
are all those sets which are not of first category.
422 JENS MARKLOF
1.15. A brief review. After its formulation in 1977, Sarnak [25] was the
first to prove the Berry-Tabor conjecture for the pair correlation of almost all
positive definite binary quadratic forms
αm
2
+ βmn + γn
2
,m,n∈
(“almost all” in the measure-theoretic sense). These values represent the eigen-
values of the Laplacian on a flat torus. His proof uses averaging techniques to
reduce the pair correlation problem to estimating the number of solutions of
systems of diophantine equations. The almost-everywhere result then follows
from a variant of the Borel-Cantelli argument. For further related examples
of sequences whose pair correlation function converges to the uniform density
almost everywhere in parameter space, see [20], [22], [30], [31], [34]. Results
on higher correlations have been obtained recently in [21], [23], [32].
Eskin, Margulis and Mozes [8] have recently given explicit diophantine
conditions under which the pair correlation function of the above binary
quadratic forms is Poisson. Their approach uses ergodic-theoretic methods
based on Ratner’s classification of measures invariant under unipotent flows.
This will also be the key ingredient in our proof for the inhomogeneous set-up.
New in the approach presented here is the application of theta sums [13], [14],
[15].


ψ
2

λ
k
λ

ˆ
h(λ
j
− λ
k
).
Here ψ
1
,ψ
2
∈S(
+
) are real-valued, and S(
+
) denotes the Schwartz class
of infinitely differentiable functions of the half line
+
(including the origin),
which, as well as their derivatives, decrease rapidly at +∞.Itishelpful to
think of ψ
1
,ψ

lim
λ→∞
R
2
(ψ
1
,ψ
2
,h,λ)=

ˆ
h(0) + π

ˆ
h(s) ds


∞
0
ψ
1
(r)ψ
2
(r) dr.
The first term comes straight from the terms j = k; the second one is the
more interesting.
Theorem 2.2 implies Theorem 1.8 by a standard approximation argument
(Section 8).
2.3. Using the Fourier transform we may write
R


λ
j
λ

e(
1
2
λ
j
u)

h(u) du.
We will show that the inner sums can be viewed as a theta sum (see 4.14 for
details)
θ
ψ
(u, λ)=
1
√
λ

j
ψ

λ
j
λ

e(

infinity (Section 6).
The only exception is a small neighbourhood of u =0,where in fact a
positive mass escapes to infinity, giving a contribution
2π
2
h(0)

∞
0
ψ
1
(r)ψ
2
(r) dr = π
2

ˆ
h(s) ds

∞
0
ψ
1
(r)ψ
2
(r) dr,
which is the second term in Theorem 2.2.
424 JENS MARKLOF
The remaining part of the orbit becomes equidistributed under the above
diophantine conditions, which yields

0
ψ
1
(r)ψ
2
(r) dr

h(u) du,
which finally yields
π
ˆ
h(0)

∞
0
ψ
1
(r)ψ
2
(r) dr,
the first term in Theorem 2.2.
The proof of Theorem 1.13, which provides a set of counterexamples to
the convergence to uniform density, is given in Section 9.
3. Schr¨odinger and Shale-Weil representation
3.1. Let ω be the standard symplectic form on
2k
, i.e.,
ω(ξ, ξ

)=x · y

2k
× with mul-
tiplication law [12]
(ξ,t)(ξ

,t

)=(ξ + ξ

,t+ t

+
1
2
ω(ξ, ξ

)).
Note that we have the decomposition

x
y

,t

=

x
0

, 0

k
,

W

0
y

, 0

f

(w)=f(w −y), with y, w ∈
k
,
W (0,t)=e(t)id, with t ∈
.
INHOMOGENEOUS QUADRATIC FORMS 425
Therefore for a general element (ξ,t)in
(
k
)

W

x
y

,t



)R(M)R(M

)
with cocycle c(M,M

) ∈ , |c(M,M

)| =1,but c(M,M

) =1in general.
3.4. For our present purpose it suffices to consider the group SL(2,
)
which is embedded in Sp(k,
)by

ab
cd

→

a 1
k
b 1
k
c 1
k
d 1
k







|a|
k/2
e(
1
2
w
2
ab)f(aw)(c =0)
|c|
−k/2

k
e

1
2
(aw
2
+ dw


2
) − w · w

c

d
1

,M
2
=

a
2
b
2
c
2
d
2

,M
3
=

a
3
b
3
c
3
d
3

∈ SL(2, ),

3.7. In the special case when
M
1
=

cos φ
1
−sin φ
1
sin φ
1
cos φ
1

,M
2
=

cos φ
2
−sin φ
2
sin φ
2
cos φ
2

,
we find
c(M

v
1/2
0
0 v
−1/2

cos φ −sin φ
sin φ cos φ

=(τ,φ),
where τ = u +iv ∈
, φ ∈ [0, 2π). This parametrization leads to the well
known action of SL(2,
)on × [0, 2π),

ab
cd

(τ,φ)=(
aτ + b
cτ + d
,φ+ arg(cτ + d)mod2π).
We will sometimes use the convenient notation (Mτ,φ
M
):=M(τ, φ) and
u
M
:= Re(Mτ), v
M
:= Im(Mτ).







f(w)(φ =0mod2π)
f(−w)(φ = π mod 2π)
|sin φ|
−k/2

k
e

1
2
(w
2
+ w


2
) cos φ − w · w

sin φ

f(w

) dw


±iπkπ/4
f(w),
and hence this projective representation is in general discontinuous at φ = νπ,
ν ∈ . This can be overcome by setting
˜
R(τ,φ)=e
−iπkσ
φ
/4
R(τ,φ).
In fact,
˜
R corresponds to a unitary representation of the double cover of
SL(2,
) [12]. This means in particular that (compare 3.7)
˜
R(i,φ)
˜
R(i,φ

)=
˜
R(i,φ+ φ

),
where φ ∈ [0, 4π) parametrizes the double cover of SO(2) ⊂ SL(2,
).
4. Theta sums
4.1. The Jacobi group is defined as the semidirect product [1]
Sp(k,


)R(M),
(recall 3.3) we have
W (ξ,t)R(M ) W (ξ

,t

)R(M

)
= W (ξ,t)W (Mξ

,t

) R(M)R(M

)
= c(M, M

)
−1
W (ξ + Mξ

,t+ t

+
1
2
ω(ξ,Mξ


y

,
Θ
f
(τ,φ; ξ,t)=v
k/4
e(t −
1
2
x · y)

m∈
k
f
φ
((m − y)v
1/2
)e(
1
2
m − y
2
u + m · x),
where
f
φ
=
˜
R(i,φ)f.

.
Proof. Since f ∈S(
k
), we can use repeated integration by parts to show
that





|sin φ|
−k/2

k
e

1
2
(w
2
+ w


2
) cos φ − w · w

sin φ

f(w


iπk
the Fourier transform of f
and therefore of Schwartz class as well. Again, after integration by parts,





|sin φ|
−k/2

k
e

1
2
(w
2
+ w


2
) cos φ − w · w

sin φ

f
π/2
(w


(w)|≤c

R
(1 + w)
−R
,
for all φ/∈ (νπ +
1
2
π −
1
100
,νπ+
1
2
π +
1
100
), ν ∈ .
Clearly for each φ ∈
at least one of the bounds applies; we put c
R
=
max{c

R
,c

R
}.

k
),

m∈
k
[Ff](m)=

m∈
k
f(m)
where F is the Fourier transform. Because
F = R(i,π/2) = R(S),S=

0 −1
10

,
and secondly
˜
R(τ,φ; ξ,t)f ∈S(
k
) for fixed (τ,φ; ξ,t), the Poisson summation
formula yields

m∈
k
[R(S)
˜
R(τ,φ; ξ,t)f](m)=


R(i, arg τ),
since (τ,0) and (−
1
τ
, 0) are upper triangular matrices, and hence the cor-
responding cocycles are trivial, i.e., equal to 1 (recall 3.6). Finally, since
0 < arg τ<πfor τ ∈
,
R(i, arg τ)
˜
R(i,φ)=e
iπk/4
˜
R(i, arg τ)
˜
R(i,φ)=e
iπk/4
˜
R(i,φ+ arg τ).
Collecting all terms, we find
R(S)
˜
R(τ,φ; ξ,t)=e
iπk/4
˜
R

−
1
τ

f

τ +1,φ;

s
0

+

11
01

x
y

,t+
1
2
s · y

=Θ
f

τ,φ;

x
y

,t



, 0

f

(m)=

m∈
k
f(m),
and hence also (replace f with
˜
R(τ,φ; ξ,t)f)

m∈
k

˜
R

i+1, 0;

s
0

, 0

˜
R(τ,φ; ξ,t)f


R

τ +1,φ;

s
0

+

11
01

x
y

,t+
1
2
s · y

,
where we have used that c((i, 0), (τ, φ)) = 1 since (i, 0) is an upper triangular
matrix; cf. 3.6.
Jacobi 3.
Θ
f

τ,φ;

k

,r

f

(m)=e(−
1
2
k · l)

m∈
k
f(m),
INHOMOGENEOUS QUADRATIC FORMS 431
and therefore, replacing f with W (ξ,t)
˜
R(τ,φ)f,

m∈
k

W

k
l

,r

W (ξ,t)
˜
R(τ,φ)f


; ξ

)=(MM

; ξ + Mξ

),
and put
Θ
f
(τ,φ; ξ)=v
k/4

m∈
k
f
φ
((m − y)v
1/2
)e(
1
2
m − y
2
u + m · x).
By virtue of Lemma 4.3 and the Iwasawa parametrization 3.8, Θ
f
Θ
g

,
with s =
t
(
1
2
,
1
2
, ,
1
2
) ∈
k
,isclosed under multiplication and inversion, and
therefore forms a subgroup of G
k
. Note also that the subgroup
N = {1}
2k
is normal in Γ
k
.
4.7. Lemma.Γ
k
is generated by the elements

0 −1
10


cd

→

ab
cd

;

abs
cds

+
2k

defines a group isomorphism. The matrices (
0 −1
10
) and (
11
01
) generate
SL(2,
), hence the lemma.
4.8. Proposition. The left action of the group Γ
k
on G
k
is properly
discontinuous. A fundamental domain of Γ

0 −1
10
) and (
11
01
) generate
SL(2,
), which explains F
SL(2, )
. Note furthermore that (
−10
0 −1
) generates
the shift φ → φ + π.
4.9. Proposition. For f,g ∈S(
k
), Θ
f
(τ,φ; ξ)Θ
g
(τ,φ; ξ) is invariant
under the left action of Γ
k
.
Proof. This follows directly from Jacobi 1–3, since the left action of the
generators from 4.7 is

τ,φ;

x

(τ,φ; ξ) → (τ, φ; ξ + m),
respectively.
We find the following uniform estimate.
4.10 Proposition. Let f,g ∈S(
k
).Forany R>1,
Θ
f

τ,φ;

x
y

Θ
g

τ,φ;

x
y

= v
k/2

m∈
k
f
φ
((m − y)v

x
y

= v
k/2
f
φ
((n − y)v
1/2
)g
φ
((n − y)v
1/2
)+O
R
(v
−R
),
uniformly for all (τ,φ; ξ) ∈ G
k
with v>
1
2
, y ∈ n +[−
1
2
,
1
2
]

v
−T/2
),
which holds uniformly for v>
1
2
, φ ∈ and y ∈ n +[−
1
2
,
1
2
]
k
,ifm = n.
Likewise for g
φ
,
|g
φ
((
˜
m − y)v
1/2
)|≤˜c
T
(1 + 
˜
m − yv
1/2

(v
−T/2
).
The following lemmas will be useful later on.
4.11 Lemma. The subgroup
Γ
θ
2k
,
where
Γ
θ
=

ab
cd

∈ SL(2, ): ab ≡ cd ≡ 0mod2

denotes the theta group, is of index three in Γ
k
.
Proof. It is well known [9] that Γ
θ
is of index three in SL(2, ) and
SL(2,
)=
2

j=0

.
4.12 Lemma.Γ
k
is of finite index in SL(2, ) (
1
2
)
2k
.
Proof. The subgroup Γ
θ
2k
⊂ Γ
k
is of finite index in SL(2, )
2k
and
thus also in SL(2,
) (
1
2
)
2k
.
434 JENS MARKLOF
4.13. Remark. Note that
SL(2,
) (
1
2

2k
.
4.14. In this paper, we will be interested in the case of quadratic forms in
twovariables, i.e., k =2. The corresponding theta sum (defined for general k
in 4.5) reads then
Θ
f
(τ,φ; ξ)=v
1/2

(m,n)∈
2
f
φ
((m − y
1
)v
1/2
, (n − y
2
)v
1/2
)
× e(
1
2
(m − y
1
)
2

ψ
2
(u, λ)=Θ
f
(τ,φ; ξ)Θ
g
(τ,φ; ξ)
with
τ = u +i
1
λ
,φ=0, ξ =
t
(0, 0,α,β),
and
f(w
1
,w
2
)=ψ
1
(w
2
1
+ w
2
2
),g(w
1
,w

Ψ
t
0
=

1 t
01

; 0

.
For t ∈
,Ψ
t
0
generates a unipotent one-parameter-subgroup of G
k
, denoted
by Ψ
0
.For any lattice Γ in G
k
,wenow define the flow Ψ
t
:Γ\G
k
→ Γ\G
k
by
right translation by Ψ

=

e
−t/2
0
0e
t/2

; 0

,
generates a one-parameter-subgroup of G
k
. The flow Φ
t
:Γ\G
k
→ Γ\G
k
defined by
Φ
t
(g):=gΦ
t
0
,
represents a lift of the classical geodesic flow on Γ\SL(2,
).
5.3. We are interested in averages of the form


) h(u) du,
which may therefore be interpreted as the average along an orbit of the unipo-
tent flow Ψ
u
, which is translated by Φ
t
. Since ρ
t
(1) = 1, ρ
t
defines a probability
measure on Γ\G
k
.
5.4. Proposition. Let Γ beasubgroup of SL(2,
)
2k
of finite index.
Then the family of probability measures {ρ
t
: t ≥ 0} is relatively compact, i.e.,
every sequence of measures contains a subsequence which converges weakly to
aprobability measure on Γ\G
k
.
Proof. Consider the function
X
R
(τ)=


Because Γ is a finite index subgroup of SL(2,
)
2k
, X
R
represents the
characteristic function of a set in Γ\G
k
, whose complement is compact.
436 JENS MARKLOF
By construction, the function X
R
is independent of φ and ξ;wecan there-
fore apply the equidistribution theorem for arcs of long closed horocycles on
Γ\
(see, e.g., [10] and [15, Cor. 5.2]), which yields for g
0
=(i, 0; ξ),
lim
t→∞
ρ
t
(X
R
)=lim
v→0

X
R
(u+iv) h(u) du =

2
= R
−1
.
Hence, given any ε>0, we find some R>1 such that
sup
t≥0
ρ
t
(X
R
) ≤ ε.
The family of ρ
t
is therefore tight, and the proposition follows from the Helly-
Prokhorov theorem [28].
5.5. Proposition. If ν is a weak limit of a subsequence of the probability
measures ρ
t
with t →∞, then ν is invariant under the action of Ψ , i.e.,
ν ◦Ψ
= ν.
Proof. Suppose {ρ
t
i
: i ∈ } is a convergent subsequence with weak
limit ν. That is, for any bounded continuous function F ,wehave
lim
i→∞
ρ

t
0
) h(u) du
=

F (g
0
Ψ
u
0
Φ
t
0
) h(u −s exp(−t)) du.
Furthermore



ρ
t
(F ◦Ψ
s
) − ρ
t
(F )



=


|ρ
t
(F ◦Ψ
s
) − ρ
t
(F )| <ε
for all t>T. Because the function
˜
F = F ◦ Ψ
s
(s is fixed) is bounded
continuous, the limit
lim
i→∞
ρ
t
i
(F ◦Ψ
s
)=ν(F ◦ Ψ
s
)
INHOMOGENEOUS QUADRATIC FORMS 437
exists, and we know from the above inequality that
|ν(F ◦ Ψ
s
) − ν(F )|≤ε
for any ε>0. Therefore ν(F ◦ Ψ
s

are linearly independent
over
.Leth beacontinuous probability density →
+
with compact
support. Then, for any bounded continuous function F on Γ\G
k
,
lim
t→∞

F ◦Φ
t
◦ Ψ
u
(g
0
) h(u) du =
1
µ(Γ\G
k
)

Γ\G
k
Fdµ
where µ is the Haar measure of G
k
.
This theorem is a special case of Shah’s more general Theorem 1.4 in

sin φ)
where χ
j
(j =1, 2, 3) is the characteristic function of the interval [s
j
,s
j
+ δ
j
].
We assume in the following that s
j
ranges over the fixed compact interval I
j
,
and that I
3
is furthermore properly contained in
+
, i.e., s
3
≥ s for some
constant
s>0. Clearly f
δ
has compact support in SL(2, ). The function
η
D
:
2k

boundary, δ
1
,δ
2
,δ
3
> 0(sufficiently small) and s
1
∈ I
1
,s
2
∈ I
2
,s
3
∈ I
3
,
lim sup
v→0

F
δ

u +iv, 0;

0
y


in such a way that
η
D
≤

j
η
C
j
,

2k



j
η
C
j
− η
D


dξ <ε.
We may therefore assume without loss of generality that η
D
(ξ)isthe charac-
teristic function of an arbitrary cube in
2k
, i.e., η


a(u +iv)+b
c(u +iv)+d
, arg(cτ + d)

η
1
(by) η
2
(dy).
In particular (with φ = 0),
v
−1/2
γ
cos φ
γ
= v
−1/2
(cu + d),v
−1/2
γ
sin φ
γ
= v
1/2
c,
u
γ
=Re
a(u +iv)+b

χ
1
(
a
c
) χ
2
(v
−1/2
(cu + d)) χ
3
(cv
1/2
) η
1
(by) η
2
(dy),
which, after being integrated against h(u)du, yields
v

γ
χ
1
(
a
c
) χ
3
(cv


u +iv, 0;

0
y

h(u)du
 δ
2
v
1/2

γ
|d|≤A|c|
1
|c|
χ
1

a
c

χ
3
(cv
1/2
) η
1
(by) η
2

0
≤|d|−1. So,
for v sufficiently small,

F
δ

u +iv, 0;

0
y

h(u)du
 δ
2
v
1/2

b,d,m∈
0<|d|≤A|c
0
+md|
1
|c
0
+ md|
χ
1

b

(b, d) are chosen as above. We have dropped
the restriction that gcd(b, d)=1.
For terms with |m| > 1, we obtain upper bounds by observing
1
|c
0
+ md|
≤
1
(|m|−1)|d|
,
and replacing the restriction imposed by χ
3
with the condition (|m|−1)|d|≤
v
−1/2
(s
3
+ δ
3
). For terms with m =0, ±1, we have
1
|c
0
+ md|
≤
A
|d|
and we replace the restriction corresponding to χ
3

440 JENS MARKLOF
which we extend to
s
1
d −
A
|d|
≤ b ≤ (s
1
+ δ
1
)d +
A
|d|
,
and for d<0,
(s
1
+ δ
1
)d −
1
c
0
+ md
≤ b ≤ s
1
d −
1
c

v
1/2

b,d,n∈
1
|nd|
η
1
(by) η
2
(dy)+O
δ,η
(v
1/2
log v),
with the summation restricted to
s
1
|d|−
A
|d|
≤±b ≤ (s
1
+ δ
1
)|d| +
A
|d|
,n|d|≤max(A, 1)v
−1/2

(x)dx,
uniformly for |d| >v
−1/4
large enough. For |d|≤v
−1/4
we use the trivial
bound

s
1
|d|−
A
|d|
≤±b≤(s
1
+δ
1
)|d|+
A
|d|
η
1
(by)=O
δ,η
(v
−1/4
),
for small enough v. Therefore

F


k
η
1
(x)dx + O
δ,η
(v
1/4
(log v)
2
),
where the last term includes all contributions from terms with |d|≤v
−1/4
.
5.10.5. We split the remaining sum over n into terms with 0 <n<v
−1/4
and terms with n ≥ v
−1/4
.Inthe first case we have, for v → 0,
nv
1/2

0<|d|n
−1
v
−1/2
(s
3
+δ
3

(dy)  n
−1
v
−1/2
(s
3
+ δ
3
).
5.10.6. We conclude
lim sup
v→0

F
δ

u +iv, 0;

0
y

h(u)du
 δ
1
δ
2
(s
3
+ δ
3

n<v
−1/4
n
−2
=
π
2
6
< ∞ and lim
v→0

n≥v
−1/4
n
−2
=0,the
lemma is proved.
5.11. Proof of Theorem 5.7.
5.11.1. By Propositions 5.4 and 5.5, we find a convergent subsequence of
ρ
t
i
with weak limit ν invariant under Ψ . Hence for any bounded continuous
function F on Γ\G
k
,
lim
i→∞
ρ
t

0
⊂ g
−1
Hg},
S(H)=

H

∈H,H

⊂H, H

=H
N(H

),
and
T
H
= π(N(H)\S(H)),
where π is the natural quotient map G
k
→ Γ\G
k
.Wedenote by ν
H
the
restriction of ν on T
H
. Then, for any g ∈ N(H)\S(H), the group g

ticular, if ν(π(S(G
k
))) = 0, then ν = µ (up to normalization).
5.11.3. Let us suppose first that there is at least one H ∈Hwith ν
H
=0,
whose projection onto the SL(2,
)-component is a closed connected subgroup
L of SL(2,
) with L = SL(2, ) (compare Appendix B). Let Λ be the projec-
tion of Γ onto its SL(2,
)-component. Since Γ ∩ H is a lattice in H,Λ∩ L
is a lattice in L.Wecan therefore construct a bounded continuous function
F (τ, φ; ξ)=F (τ,φ) such that

Fdν =
1
µ(Γ\G
k
)

F dµ.
With F independent of ξ,weapply the equidistribution theorem for long arcs
of closed horocycles [10], [15], which yields
lim
t→∞
ρ
t
(F )=
1

= SL(2, ) Ω,
where Ω is a closed connected subgroup of
2k
(i.e., Ω is a closed linear subspace
of
2k
), which is invariant under the action of SL(2, ). Since SL(2, ) {0}
and {1}
Ω are generated by unipotent one-parameter subgroups, the same
holds for H
0
and hence for H. The right action of H on Γ\ΓH is obviously
ergodic with respect to the (unique) H-invariant probability measure ι, and
therefore H ∈H.