LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION OF
RANDOM WALKS ON THE HEISENBERG GROUP
E.BREUILLARD
To the memory of Martine Babillot
Abstract.We prove local limit theorems for products of independent random
variables on the Heisenberg group which are identically distributed with respect
to an arbitrary centered and compactly supported probability measure .We also
provide uniform estimates for translates of a bounded set by comparing
n
to the
associated heat kernel.This,in turn,enables us to show the equidistribution of
Heisenbergunipotent random walks on nite volume homogeneous spaces G=.
The goal of this paper is to show the local limit theorem for centered probability
measures on the Heisenberg group along with some of its renements and applica
tions.The local limit problem on noncommutative Lie groups has been studied by
many authors in the last thirty or forty years (ItoKawada,ArnoldKrylov,Kazhdan,
Bougerol,Le Page,Guivarc'h,Varopoulos,etc.).In the classical commutative case or
in the compact group case the local limit theorem is available under the weakest as
sumptions on the probability measure (see [Sto] and [ItK]).In [Bou],Bougerol solves
the local limit problem for absolutely continuous probability measures on semisimple
Lie groups.In a recent work,Alexopoulos [Ale] obtains a very precise local limit
theorem and estimates a la BerryEssen for probability measures with a continu
ous density of compact support on an arbitrary connected Lie group of polynomial
growth.
However,in this problem,an assumption of absolute continuity of the probability
measure with respect to the Haar measure is often made,while the case of a possibly
singular (e.g.nitely supported) measure remains generally open.Such cases include
nitely supported probability measures on the group of isometries of the Euclidean
3space (see [Kaz],[Gui2]) or the case of semisimple groups [Bou].Similarly the
speed of convergence to equidistribution is not well understood and seems to depend
on dicult arithmetic questions (cf.the spectral gap conjecture [Sar] about the
equidistribution on the sphere).In this paper however,we treat the case of an
arbitrary,possibly singular,measure.We will focus on the simplest nilpotent Lie
group:the rst Heisenberg group.
Comparing the convolution powers of the measure with the associated heat kernel,
we also obtain a uniform version of the local limit theorem yielding a fairly precise
estimate on the asymptotic behavior of centered random walks on the Heisenberg
1
2 E.BREUILLARD
group.This generalizes a result of Stone [Sto] in the commutative case.This estimate
allows to show further equidistribution results for random walks on homogeneous
spaces.Following this strategy,and making use of Ratner's theorem on orbits of
unipotent ows (see [Rat],[Sha],[Sta]),we show at the end of the paper that centered
Heisenbergunipotent randomwalks on homogeneous spaces G=;where is discrete
of nite covolume in a Lie group G;are equidistributed in the closure of the orbit
on which they live.
In the noncentered case,an interesting phenomenon can occur:noncentered
unipotent random walks may not converge to any probability measure on G=.For
instance if is not cocompact,they may stay outside an arbitrary compact set with
high probability at arbitrary large times.Hence the hypothesis that the walk should
be centered is crucial (unless of course the Haar measure is uniquely ergodic for the
unipotent subgroup).
Finally let us remark that the results of this paper should extend to the case of
an arbitrary simply connected nilpotent Lie group,but the technical diculty of the
forthcoming proofs forced us to restrict our attention to the Heisenberg group.
1.Statement of the results
Let G be the group of 33 uppertriangular unipotent matrices and e the identity
in G.Let us x the Haar measure on G,dg = dxdydz where
g =
0@
1 x z
0 1 y
0 0 1
1A
is simply denoted by g = (x;y;z).We also denote by jAj the Haar measure of a
Borel set A and we x a homogeneous norm kgk = maxfjxj;jyj;jzj
1=2
g on G:
We consider a probability measure on G with the following properties:
compactly supported.
centered:
R
p(x)d(x) = 0 where p:G!G=[G;G] is the canonical map.
aperiodic:for any proper closed subgroup H G and any x 2 G,(xH) < 1.
In particular,we make no assumption of smoothness for ,which can be for in
stance nitely supported.
The convolution product of measures is denoted by and convolution powers
simply denoted by
n
.
The central limit theorem for G is well known (see the work of Wehn [Weh],
as well as Tutubalin [Tut] and CrepelRaugi [Rau]).It states that if (d
t
)
t
is the
semigroup of dilations given by d
t
(x;y;z) = (tx;ty;t
2
z) then the sequence d 1
p
n
(
n
)
converges to some gaussian measure on G (in the sense of probability measures,
i.e.
R
f d 1
p
n
d
n
!
R
fd for every bounded continuous function on G).The
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 3
measure lies inside a gaussian semigroup of probability measures (
t
)
t>0
dened by
its generating distribution
(1) Af =
d
dt
t=0
Z
f(x)d
t
(x) =
z@
z
f(e) +
xy@
2
xy
f(e) +
1
2
x
2
@
2
x
f(e) +
1
2
y
2
@
2
y
f(e)
where
z =
R
zd(x;y;z) and
xy =
R
xyd(x;y;z).Then =
1
and
t
= p
t
(g)dg
where p
t
(g) is the heat kernel associated to the operator dened by (1),hence is a
strictly positive fastly decreasing smooth function on G (see [VSC]).It is the density
of the Brownian Motion corresponding to A on G.
Let c() = p
1
(e) > 0.For general references about gaussian semigroups and the
LevyKhintchinHunt formula,see [Gre],[Neu] and the original article of Hunt [Hun],
as well as the survey article [Breu2] where a selfcontained proof of Wehn's central
limit theorem can be found.
We say that satises Cramer's condition if
sup
t
2
+s
2
1
Z
e
i(tx+sy)
d(x;y;z)
< 1
We obtain the theorems below without this assumption,but at some point we get a
slightly stronger result if this assumption is made (apparently for technical reasons,
but we have no guess whether it is necessary).
Let us state the local limit theorem for G together with a uniform version for
translates of a bounded set.These results generalize to the Heisenberg group the
well known theorems of classical probability theory on R
n
(see [Bre] and [Sto]).
The method of proof makes use of the unitary representations of G to obtain a
crucial spectral bound (Proposition 3.2).The main point after that is the study of
an auxiliary quadratic dynamical system (see equations (30)),in order to obtain a
key estimate similar to the domination condition appearing in the classical proof of
the local limit theorem on the real line (see [Bre]).
Theorem 1.1.(Local limit theorem) Let be a compactly supported aperiodic cen
tered probability measure on G.Let f be a compactly supported continuous function
on G.Then the following convergence holds uniformly when z varies in compact
subsets of G
lim
n!+1
n
2
Z
G
f(gz)d
n
(g) = c()
Z
G
f(g)dg
Theorem 1.2.(Uniform local limit theorem) Let be a compactly supported aperi
odic centered probability measure on G and (v
t
)
t>0
the corresponding limit gaussian
semigroup.Then for any bounded Borel subset B G with j@Bj = 0,
lim
n!+1
sup
x2G
n
2
j
n
(xB)
n
(xB)j = 0
4 E.BREUILLARD
If we assume additionally Cramer's condition,then
lim
n!+1
sup
x;y2G
n
2
j
n
(xBy)
n
(xBy)j = 0
Let us remark that the choice of (
t
)
t>0
depends on the choice we made of a semi
group of dilations (d
t
)
t>0
.Any other choice (d
0t
)
t>0
for the semigroup of dilations
is of the form d
t
1
for some automorphism of G.The associated gaussian
semigroup (
0
t
)
t>0
is obtained from (
t
)
t>0
by composing by some automorphism of
G and Theorem 1.2 remains valid if we take (
0
t
)
t>0
instead of (
t
)
t>0
.
Theorem 1.3.(Concentration function) Under the assumptions above for ,for
every bounded set K G,there is a constant C
K
such that for all integers n
sup
x2G
n
(xK)
C
K
n
2
If we suppose additionally Cramer's condition,then
sup
x;y2G
n
(xKy)
C
K
n
2
This uniform version of the local limit theorem for translates enables to show the
following corollary.The point of this result is that the function f is not assumed to
tend to 0 at innity,and in particular,can have a periodic type of behavior.For the
classical commutative case see [Breu].
Corollary 1.4.Let f be an arbitrary leftuniformly continuous function on G such
that the following limit exists
(2) lim
T
x
!1;T
y
!1;T
z
!1
1
T
x
T
y
T
z
Z
T
x
0
Z
T
y
0
Z
T
z
0
f(x;y;z)dxdydz =`
where`2 C.Then
lim
n!+1
Z
G
f(g)d
n
(g) =`
This corollary enables to prove further equidistribution results on homogeneous
spaces.In the last section we show how to derive the equidistribution of Heisenberg
unipotent random walks on homogeneous spaces,such as horospheric random walks
on complex hyperbolic manifolds,namely,
Theorem1.5.Let G be a connected Lie group and a lattice in G.Let H be a closed
subgroup of G consisting of unipotent elements and isomorphic to the Heisenberg
group.Let be a centered compactly supported aperiodic probability measure on H.
Then for an arbitrary x 2 G= and for any bounded and continuous function f on
G=,
lim
n!+1
Z
H
f(hx)d
n
(h) =
Z
G
f(g)dm
x
(g)
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 5
where m
x
is the unique Hinvariant ergodic probability measure on G= whose support
is the closure of the orbit
Hx.
The existence of the measure m
x
is given by Ratner's theorem (see [Rat1 to 3],
[Sta]).The corresponding deterministic result was proved for one parameter sub
groups by Ratner [Rat] elaborating on a weaker qualitative recurrence result due to
Margulis [Mar] and subsequently generalized by Dani [Dan],and for general unipo
tent groups by Shah [Sha].
Note that this also implies that the only stationary probability measures on G=
(i.e.the measures such that = ) are the Hinvariant ones.This means,in
the terminology of Furstenberg (cf.[Fus]),that the action of H on G= is sti with
respect to .But this also follows from the fact that (H;) has the ChoquetDeny
property (i.e.the absence of bounded harmonic functions) as follows fromthe work
of Guivarc'h [Gui].
Also note that according to a general random ergodic theorem of Oseledec (cf.
[Ose]),which is proved in the case when is symmetric (i.e.(A) = (A
1
)),
the convergence of Theorem 1.5 above holds for malmost every x in G= for any
Hergodic probability measure m on G=.We underline that,in Theorem 1.5,we
capture the behavior of the random walk for every starting point x.
In the above theorem,the assumption that is centered cannot be removed.As
will be shown in x10.2.1,a simple use of the central limit theorem shows that for
certain (in fact almost all) lattices in R
2
;any non centered unipotent random walk
starting at that point in the space of lattices SL
2
(R)=SL
2
(Z) will diverge,i.e.may
remain very far with high probability at some arbitrary large time.
When H is uniquely ergodic on G= (e.g.the horocycle owon a compact Riemann
surface),then Theorem 1.5 follows easily from an equidistribution theorem due to
Guivarc'h (Theoreme V.5.in [Gui]) and holds even when is not centered.
This application was originally motivated by the work of Eskin and Margulis [EsM],
where they studied the case of random walks on G= obtained by a measure whose
support is Zariski dense in a semisimple group.Their main result is that the sequence
n
x
is relatively compact in the space of probability measures on G=.
2.Notations and outline of the paper
We keep the notations and terminology introduced in the last section.Let be the
regular representation of G on the functions on G:(g)f(x) = f(g
1
x).We denote
by h;i the scalar product on L
2
(G).
G is identied with its Lie algebra g by writing g = (x;y;z) with the help of the
dieomorphism
g!G(3)
xX +yY +zZ 7!e
yY
e
xX
e
zZ
where X;Y and Z are the upper triangular elementary matrices,with [X;Y ] = Z.
6 E.BREUILLARD
The product in G is given by
gg
0
= (x +x
0
;y +y
0
;z +z
0
+xy
0
)
In the sequel,we will need to look at possible other parametrizations of G,in
particular at those of the form
:R
3
!G
(x;y;z) 7!(x;y;z +(x;y))
where (x;y) is a quadratic form in x and y.The Fourier transform of a function
on G can be dened in dierent ways depending on the choice of a parametrization.
Given a function f:G!C,we shall denote by F
(f):(R
3
)
_
!C the Fourier
transform taken in the parametrization dened by
.When = 0,we simply write
F
0
(f) =
b
f.The variable in the dual space (R
3
)
_
will be denoted by = (t;s;).The
formula reads:
F
(f):(R
3
)
_
!C(4)
= (t;s;) 7!
1
(2)
3=2
Z
f(
(x;y;z))e
i(tx+sy+z)
dxdydz
Let C
c
(G) be the space of continuous and compactly supported functions on G.
Let be a probability measure satisfying the properties listed in the introduction
and (X;Y;Z) a random variable on G distributed according to .We set once and
for all
(5) =
E(XY )
2E(Y
2
)
Theorem 2.1.Let (x;y) = y
2
and F
the Fourier transform just dened.Let f
and g be two functions on G with f 2 C
c
(G),g integrable and F
(g) 2 C
c
(G).Then
lim
n!+1
n
2
h(
n
)f;gi = c()
Z
G
f
Z
G
g
and,if we suppose additionally that F
(g) is absolutely continuous,then uniformly
when z varies in compact subsets,
lim
n!+1
n
2
Z
g(z
1
x)d
n
(x) = c()
Z
G
g
Below,we derive Theorem 1.1 from Theorem 2.1.The strategy for proving Theo
rem2.1 follows the general scheme provided by Stone's proof of the local limit theorem
on R
d
[Sto] and is as follows.Looking at the Fourier transform of the integral,we
give an explicit decomposition of the regular representation of G into a continuous
direct sum of primary representations and treat each part of the integral to show that
only the part with small 's and small s and t's gives a contribution.Then we gain
control on this part by showing a domination condition on the integrand.This is
achieved by performing a Taylor expansion in s;t; of the characteristic function of
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 7
n
.Lebesgue's dominated convergence theorem combined with the pointwise con
vergence granted by the central limit theoremon Gcompletes the proof.The proof of
Theorem (1:2) makes use of the estimates previously obtained and goes along similar
lines.
3.Irreducible unitary representations of G
The irreducible unitary representations of G are well known.Apart from char
acters,there is a oneparameter family of irreducible unitary representations
( 2 Rnf0g) modeled on L
2
(R) by
(g)f(t) = e
i(tx+z)
f(t +y)
The following two propositions will be crucial in the proof.The rst is quite
standard (see [Gre],and [Gui]):
Proposition 3.1.Let be an aperiodic probability measure on G.Then for any
closed interval I Rnf0g
(6) sup
2I
k
()k < 1
Proof.Let
n
! 2 I be such that
n
(
1
)f
n
(t);f
n
(t)
!1
for some sequence of vectors f
n
2 L
2
(R) of norm1.Then,up to taking a subsequence,
for
1
almost every x,
h
n
(x)f
n
(t);f
n
(t)i!1
But = fx 2 G;h
n
(x)f
n
(t);f
n
(t)i!1g is clearly a subgroup of G,and
1
() = 1.Since is aperiodic, is dense in G,hence [;] is dense in the
center of G.In particular,we can nd (0;0;z) 2 ,such that z =2 2Z.Then
e
i
n
z
f
n
f
n
!0
which implies e
iz
= 1 and provides the desired contradiction.
The next proposition gives an estimate of the norm of the operators
().When
is taken to be the symmetric Dirac measure on (1;0;0) and (0;1;0),this operator
can be viewed as acting on`
2
(Z) and then coincide with the wellknown Harper
operator (see [BVZ]) studied in mathematical physics.
Proposition 3.2.Let be a probability measure on G whose support is not contained
in a coset of an abelian subgroup of G.Then we have
(7) lim inf
!0
1 k
()k
jj
> 0
8 E.BREUILLARD
Proof.Note that if we dene
1
(B) = (B
1
) for every Borel subset B of G,then
(
1
) is the adjoint of
() and
(
1
) is selfadjoint and nonnegative.
If (7) does not hold,then we can nd unit vectors f
2 L
2
(R) such that for
arbitrarily small 's
(
1
)f
;f
1 jo()j
From the assumption made on ,we can nd two noncommuting elements x
0
and
x
1
lying in the support of
1
.For each we can then nd x
0
close to x
0
and x
1
close to x
1
(i.e.
x
1
x
1
and
x
0
x
0
< kx
1
x
0
k=3) such that for i = 0;1
Re
(x
i
)f
;f
1 jo()j
The commutator (x
0
;x
1
) belongs to the center,hence is of the form (0;0;c
).From
the choice of x
i
it follows that c < c
< 1=c for some constant c 2 (0;1).From the
StoneVon Neumann theorem (cf.[CoG]) we can then nd an isometry I
of L
2
(R)
such that,conjugating by I
,
(x
0
) is turned into the translation by c
and
(x
1
)
is turned into the multiplication by e
it
.Hence we can assume that for arbitrarily
small 's
Re hf
(t +c
);f
(t)i 1 jo()j
Re
e
it
f
(t);f
(t)
1 jo()j
Or equivalently
kf
(t +c
) f
(t)k = o(
p
jj)(8)
e
it
f
(t) f
(t)
= o(
p
jj)(9)
Let A
"
= ft 2 R;d(t;2Z) <"g.We deduce from (9) that
(10)
Z
A
c"
jf
(t)j
2
dt = o(jj="
2
)
and from (8) that for any positive integer n
kf
(t +nc
) f
(t)k = o(n
p
jj)
or(11) Re hf
(t +nc
);f
(t)i 1 o(n
2
)
We now take n = [
1
2
p
jj
] +1 and"= c
p
jj=12.Then for small enough we have
1 > jnc
j > 3".So for small and as soon as"< ( 1)=2,making use of (10)
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 9
and applying the CauchySchwarz inequality we have
jhf
(t +nc
);f
(t)ij
s
Z
A
"
jf
(t +nc
)j
2
Z
A
"
jf
(t)j
2
+
s
Z
A
c"
jf
(t +nc
)j
2
Z
A
c"
jf
(t)j
2
2
s
Z
A
c"
jf
(t)j
2
= o(
p
jj=") = o(1)
which yields the desired contradiction with (11).
4.Reducing to small values of
Here we will begin the proof of Theorem (2:1).Recall that G is identied with its
Lie algebra via the choice of the basis (X;Y;Z) and the coordinates (x;y;z) dened
as in equation (3).The center of G is the one parameter subgroup H = e
RZ
.We
are going to decompose the regular representation of G into a continuous direct sum
of other unitary representations.Every character of H is determined by a number
2
b
H
=
R
_
.If f 2 L
1
(G);we dene for x 2 G
f
(x) =
1
p
2
Z
R
f(xe
zZ
)e
iz
dz
We check that f
(xe
zZ
) = e
iz
f
(x).By the Fourier isometry,if z 7!f(xe
zZ
) is in
L
2
(R),then 7!f
(x) is in L
2
(R
_
),and
Z
R
jf(xe
zZ
)j
2
dz =
Z
R
_
jf
(x)j
2
d
By Fubini's theorem,it follows that,if f also belongs to L
2
(G) then jf
(x)j is in
L
2
(G=H) for almost every .Let us write H
the Hilbert space of measurable
functions F on G such that F(xe
zZ
) = e
iz
F(x) and jF(x)j is square integrable on
G=H.Then H
is a realization of the induced representation
= Ind
GH
,where G
acts by left translations.The above Plancherel formula for f
shows that we have
the continuous sum decomposition
=
Z
d
and if f;g belong to L
2
(G)
h(
n
)f;gi =
Z
h
(
n
)f
;g
i
H
d
10 E.BREUILLARD
It is easy to see that
is a primary unitary representation of G and that the
representation
dened above is the only irreducible representation of G contained
in
:Moreover,its multiplicity is innite:
(12)
=
Z
(s)ds
with
(s)'
for all s.
Now from Proposition (3:2) and from (12) we obtain that for some"2 (0;1) there
exists some c > 0 such that for all 2 R with jj "and all n 2 N
(13) k
(
n
)k k
()k
n
e
cjjn
therefore for any integer k
0
3,whenever D > k
0
=c we have
n
2
Z
D
log n
n
jj"
h
(
n
)f
;g
i
H
d
n
2
Z
D
log n
n
jj"
e
k
0
log n
kf
k
H
kg
k
H
d
1
n
k
0
2
Z
2R
kf
k
H
kg
k
H
d
1
n
k
0
2
kfk
L
2
(G)
kgk
L
2
(G)
!0 as n!1(14)
The last step follows from the CauchySchwarz inequality.Hence this part of the
integral tends to 0.Similarly if I denotes any of the intervals [A;"] or [";A]
where A is some positive number such that g
is identically zero outside [A;A],
then it follows from Proposition (3:2) and from (12) that there is some constant
2 (0;1) such that
sup
2I
k
(
n
)k
n
Hence
n
2
Z
2I
h
(
n
)f
;g
i d
n
2
n
Z
2R
kf
k
H
kg
k
H
d
n
2
n
kfk
L
2
(G)
kgk
L
2
(G)
C
n
k
0
2
kfk
L
2
(G)
kgk
L
2
(G)
(15)for some constant C > 0 depending on .The right hand side clearly tends to 0 as
n tends to innity.
Now observe that if F
(g) (dened in Theorem 2.1) has compact support,then
g
is identically zero outside some bounded set of values of .More precisely,if for
some xed ,F
(g)(t;s;) = 0 for all t and s,then g
vanishes identically.The last
argument allows then to reduce to small values of (i.e.less that D
log n
n
for some
xed D > 0).
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 11
We are now going to perform Fourier integration one step further,i.e.on G=H.
We x an arbitrary Borel section
:G=H!G (given in the above coordinates by
(x;y) 7!(x;y;(x;y)) where is some measurable function on R
2
).Then f
(
(
x))
is in L
2
(G=H) for almost every .
In those coordinates,we can write down explicitely the decomposition of into the
continuous sum of the
's.Indeed,let f and g be two functions in L
1
(G)\L
2
(G)
and y be given in G.We have on the one hand:
h(y)f;gi =
Z
R
_
(y
1
)f
;g
H
d
and on the other hand,
h(y)f;gi =
Z
G=H
Z
H
f(y
1
xe
zZ
)
g(xe
zZ
)dzd
x
=
Z
G=H
Z
R
_
f
(y
1
x)
g
(x)dd
x
=
Z
R
_
Z
G=H
f
(y
1
x)
g
(x)d
xd
=
Z
R
_
Z
R
2
f
(y
1
(x;y))
g
(
(x;y))dxdyd
For notational convenience we set
y;;
(x;y):= f
(y
1
(x;y)) and
;
(x;y):=
g
(
(x;y)):For almost all 2 R,these functions belong to L
2
(R
2
).Performing
Fourier transform on L
2
(R
2
) now we get:
h(y)f;gi =
Z
R
_
Z
R
2_
[
y;;
(t;s)
d
;
(t;s)dtdsd
Now a straightforward computation yields that
d
;
(t;s) is the Fourier transform at
(t;s;) of g
dened by
(16) g
:= (x;y;z) 7!g(x;y;(x;y) +z)
d
;
(t;s) =
b
g
(t;s;)
Similarly if y = (y
x
;y
y
;y
z
) we compute,
[
y;;
(t;s) =
1
(2)
3=2
Z
e
i(y
x
(t+y)+y
y
s+(y
z
(x+y
x
;y+y
y
)
e
i(tx+sy+z)
f(x;y;z)dxdydz
Hence,if we suppose additionally that f has compact support on G and F
(g) =
b
g
has compact support on R
3
,we obtain
h(y)f;gi =
1
(2)
3=2
Z
dxf(x)
Z
R
_
Z
R
2_
e
i[y
x
(t+y)+y
y
s+(y
z
(x+y
x
;y+y
y
))]
e
i(tx+sy+z)
b
g
(t;s;)d
12 E.BREUILLARD
in other words,
h(y)f;gi =
Z
G
dx
(2)
3=2
f(x)
e
i (;y;x)
;
b
g
(t y;s;)
=(t;s;)2R
3
where x = (x;y;z),y = (y
x
;y
y
;y
z
), = (t;s;) and (;y;x) = t(y
x
+x) +s(y
y
+
y) +(z xy +y
z
(x +y
x
;y +y
y
)).
Let S is a random variable in G with distribution
n
.The quantity we are inter
ested in is
n
2
E(h(S)f;gi) =
Z
G
dx
(2)
3=2
f(x)
E(e
i (;S;x)
);
b
g
(t y;s;)
=(t;s;)2R
3
Let D > 0.We split it in two parts and write n
2
E(h(S)f;gi) = A
n
+B
n
where
A
n
=
Z
G
dx
(2)
3=2
f(x)J
n
(x)
B
n
=
Z
G
dx
(2)
3=2
f(x)I
n
(x)
and
J
n
(x) = n
2
Z
jjD
log n
n
E(e
i (;S;x)
);
b
g
(t y;s;)
(t;s)2R
2
d
I
n
(x) = n
2
Z
jjD
log n
n
E(e
i (;S;x)
);
b
g
(t y;s;)
(t;s)2R
2
d
The part A
n
has already been dealt with,because the above computations show that
A
n
= n
2
Z
jjD
log n
n
h
(
n
)f
;g
i
H
d
and (applying (14) and (15)) there is C 0 (depending on D, and the size of the
set f;9(t;s);F
g
(g)(s;t;) 6= 0g) such that if n 1
(17) jA
n
j
C
n
k
0
2
kfk
L
2
(G)
kgk
L
2
(G)
which tends to zero as soon as k
0
is taken such that D k
0
=c 3=c (where c was
the constant dened in (13)).
Hence in the sequel,xing x = (x;y;z) 2 G;we shall focus on the term I
n
(x).
Before going further,we shall x once and for all the section .We take it of the
form proposed in Theorem 2.1,that is (x;y) = y
2
where is dened in terms of
the moments of in equation (5):In this case,
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 13
E(e
i (;S;x)
) = E(e
i[t(S
x
+x)+s(S
y
+y)+(z+S
z
(x+S
x
;y+S
y
))]
)
= e
i(tx+sy+z)
E(e
i
[
tS
x
+sS
y
+S
z
(y+S
y
)
2
]
)
= e
i(tx+sy+(zy
2
))
E(e
i
[
tS
x
+(s2y)S
y
+(S
z
S
2
y
)
]
Hence,
I
n
(x) = n
2
Z
jjD
log n
n
E(
n
(t;s;));e
i
b
g
(t y;s +2y;)
(t;s)2R
2
)
d
where = tx +sy +(z xy +y
2
) and
(18)
n
(t;s;) = e
i
[
tS
x
+sS
y
+(S
z
S
2
y
)
]
We shall next estimate E(
n
(t;s;)) for small values of .
5.Evaluation of the integral for small
Let us x D > 0.The remainder of this section is devoted to nding a suitable
bound for the expectation E(
n
(t;s;)) when jj D
log n
n
and s and t take values
that are bounded away from 0 and innity.
Let G
n
= (X
n
;Y
n
;Z
n
) be a sequence of independent random variables identically
distributed according to the probability measure on G.We write S
n
= G
n
::: G
1
the product of these variables.The law of S
n
is
n
.Bearing in mind the form of the
product on G,we get S
n
= (S
n;x
;S
n;y
;S
n;z
) where
S
n;x
= X
1
+:::+X
n
S
n;y
= Y
1
+:::+Y
n
S
n;z
= Z
1
+:::+Z
n
+X
2
Y
1
+X
3
(Y
1
+Y
2
) +::::+X
n
(Y
1
+:::+Y
n1
)
Let
n
be the random variable dened in (18) as follows
n
=
n
(t;s;) = e
i(tS
n;x
+sS
n;y
+(S
n;z
S
2
n;y
))
(19)
= e
iU
2
n
Y
1kn
e
iX
k
(t+U
k1
)
e
i(sY
k
+Z
k
)
and
0
= 1
where U
k
= Y
1
+:::+Y
k
and U
0
= 0 as above.In the sequel we x the value of (as
in equation (5)) to be
= E(XY )=2E(Y
2
):
We will also use the following notation:
(t;s) = E(e
i(X
1
t+Y
1
s)
)
14 E.BREUILLARD
5.1.Moderate deviations.We shall need the following lemma about moderate
deviations.Since we could not nd a reference for this precise form of the estimate
we want (about the maximum of the random walk up to time n),we include a proof.
It follows the well known argument of Cramer via Laplace transforms.
Lemma 5.1.Let U
0
= 0 and U
n
= Y
1
+:::+Y
n
be a sum of independent identically
distributed real random variables Y
n
's,which are assumed of compact support and
centered.Let A
n
be the event fmax
0kn
jU
k
j
p
nlog ng and let A
cn
be the com
plementary event.Then for every nonnegative integer p;there are constants c
p
> 0
and C
p
> 0 such that for all integers n 2 N
E( max
0kn
jU
k
j
p
1
A
cn
) C
p
e
c
p
log
2
n
Proof.Let Y be some random variable distributed according to the common distri
bution of the Y
n
's.Since Y has compact support,if D > 0 is a bound for the support,
we obviously get
jU
n
j Dn
for all n.Hence for any xed p 0
E( max
0kn
jU
k
j
p
1
A
cn
) D
p
n
p
P(A
cn
)
Therefore it is enough to show the lemma when p = 0.
We can assume that Y is not identically 0.Dene the Laplace transform () of
Y by e
()
= E(e
Y
) for a positive real > 0.Then dene the Fenchel transform
(x) = sup
>0
(x ()) for a given x > 0.Clearly,
(x) is a nondecreasing
function of x > 0.The function () is strictly convex,since its second derivative
00
() = e
2()
(E(e
Y
)E(Y
2
e
Y
) E(Y e
Y
)
2
) is > 0 from the CauchySchwarz in
equality.In particular the supremum
(x) is attained for a unique value
x
of
given by the equation
0
(
x
) = x.And if x > 0 then
x
> 0 because
0
(0) = 0 since
Y is centered:Dierentiating the relation
0
(
x
) = x,we obtain
d
x
dx
=
1
00
(
x
)
hence
d
dx
=
x
and
d
2
dx
2
=
1
00
(
x
)
> 0
Therefore
(x) is strictly convex for x > 0:
Since Y has compact support,it has moments of any order and in particular we
have the following Taylor expansion
() =
2
2
E(Y
2
) +O(
3
)
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 15
and(20)
0
() = E(Y
2
) +O(
2
)
In particular,as x tends to 0 the value
x
tends to 0 too.We set
(0) = 0:By
denition of
,we have for any x > x
0
> 0
(21)
(x)
x
0
x (
x
0
)
Now let us study the function
near 0.From (20),we get
x
=
x
E(Y
2
) +O(
x
)
and
lim
x!0
+
x
E(Y
2
)
x
= 1
Hence
(x) =
x
x (
x
)
=
x
x
2x
E(Y
2
)
2
+O(
3x
)
=
x
2
E(Y
2
)
(u
u
2
2
) +O(ux)
3
where u =
x
E(Y
2
)=x.As x tends to 0,u tends to 1,therefore
(22)
(x)
x
2
4E(Y
2
)
for any suciently small x.
Similarly
(
x
)
x
=
2x
2x
E(Y
2
) +O(
3x
x
)
=
x
u
2
+O(
2x
u)
Hence(23)
(
x
)
x
3
4
x
for all suciently small x:Fix x
0
> 0 such that both (22) and (23) hold for all
x 2 (0;x
0
].From (21) we obtain immediately for all x x
0
(24)
(x)
x
0
(x
3
4
x
0
)
Now let us now apply these estimates obtained above for
(x) to the probability
P(U
k
>
p
nlog(n)).We write for x > 0 and > 0
E(e
U
n
) e
x
P(U
n
> x)
16 E.BREUILLARD
or
P(U
n
> x) e
n(
x
n
())
Hence taking the supremum of > 0
(25) P(U
n
> x) exp(n
(
x
n
))
Take two integers k and n with k n.We have
P(U
k
>
p
nlog(n)) exp(k
(
p
nlog n
k
))
Suppose rst that
p
nlog n
k
x
0
.Then it follows from (24) that
P(U
k
>
p
nlog(n)) exp(
x
0
p
nlog n +
3
4
kx
0
x
0
)
exp(
1
4
x
0
p
nlog n) e
c(log n)
2
where we can take c =
x
0
=4.
Now assume that
p
nlog n
k
x
0
.Then it follows from (22) that
P(U
k
>
p
nlog(n)) e
c(log n)
2
where c can be taken to be 1=4E(Y
2
).
By taking the lesser of the two c above we can now write
P(U
k
>
p
nlog(n)) e
c(log n)
2
for all integers n and k with n k 1:Hence
P(A
cn
) 2ne
c(log n)
2
Therefore there is a constant c
0
> 0 smaller that c such that when n is larger than
say n
0
we have
P(A
cn
) e
c
0
(log n)
2
For C
0
we may take e
c
0
(log n
0
)
2
and we have obtained the desired inequality.
Obviously,we may,and do,assume that c
p+1
< c
p
for all p.This lemma will enable
us to reduce to the case when U
k
does not take very big values.Let us also remark
that in the above proof,the constant c
p
can be chosen to depend only on the size of
the support of Y,i.e.on M = minft;P(jY j < t) = 1g.In the sequel,we will use
freely the result of Lemma 5.1,in particular the fact that E(jU
n
j
p
) = O
p
(n
p=2
log
p
(n))
for any p 0.
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 17
5.2.Estimate for\large"values of s and t.
Proposition 5.2.Let";A;D be positive numbers with A >"> 0.Let k be an
arbitrary positive integer.Then the following estimate holds uniformly when 2
[D
log n
n
;D
log n
n
] and"< s
2
+t
2
< A:
E(
n
) = O(
1
n
k
)
where the constant in O depends on";A;k;D and .
Proof.The proof will proceed by induction.Recall that A
n
is the event fmax
0kn
jU
k
j
p
nlog ng.In fact,we will take for induction hypothesis the following statement,
which we denote by H
j;k
for j 2 N and k 2 Z:
E(
n
U
j
n
) = O(
log
2k+3j
n
n
k=2
)
holds uniformly when the parameters (s;t;) vary in the range dened in the state
ment of the proposition,and where O depends only on ,",A,D,k and j.
Clearly H
0;k
for all k 0 implies the proposition.Also note that H
j;k
implies
H
j;k1
.
Let k + j 0.Suppose that D > 0 is given and let 2 [D
log n
n
;D
log n
n
].We
do not make an assumption on (s;t) for the moment.Making use of Lemma 5.1
and of the independence of the random variables G
i
's,we have the following Taylor
expansion.For all positive integers p n,
E(
p
U
j
p
) = E(
p
U
j
p
1
A
p
) +O
;j
(e
c
j
log
2
p
)
= E(
p1
U
j
p
e
i(2U
p1
Y
p
+Y
2
p
)
e
iX
p
(t+U
p1
)
e
i(sY
p
+Z
p
)
1
A
p
) +O
;j
(e
c
j
log
2
p
)
= E[
p1
U
j
p
e
i(X
p
t+Y
p
s)
1
A
p
k+j
X
l=0
(i)
l
l!
l
X
q=0
C
q
l
(Z
p
Y
2
p
)
lq
((X
p
2Y
p
)U
p1
)
q
]+
+
k+j+1
O
;D;k;j
((
p
plog p)
k+2j+1
) +O
;j
(e
c
j
log
2
p
)
18 E.BREUILLARD
Further expanding and using Lemma 5.1 again:
E(
p
U
j
p
) = E[
p1
j
X
r=0
C
r
j
U
r
p1
Y
jr
p
!
e
i(X
p
t+Y
p
s)
k+j
X
l=0
(i)
l
l!
l
X
q=0
C
q
l
(Z
p
Y
2
p
)
lq
((X
p
2Y
p
)U
p1
)
q
] +O
;D;k;j
(
log
2k+3j+2
p
p
(k+1)=2
)
=
k+j
X
l=0
l
X
q=0
j
X
r=0
C
r
j
C
q
l
(i)
l
l!
E(
p1
U
r+q
p1
)
E(e
i(X
1
t+Y
1
s)
Y
jr
1
(Z
1
Y
2
1
)
lq
(X
1
2Y
1
)
q
)+
+O
;D;k;j
(
log
2(k+1)+3j
p
p
(k+1)=2
)
The last expression can be written as a sum of (t;s)E(
p1
U
j
p1
) (where (t;s) =
E(e
i(X
1
t+Y
1
s)
)) and a linear combination with bounded coecients of a bounded num
ber of terms (the bounds depend only on ,D,k and j) of the form
l
E(
p1
U
m
p1
)
with 0 m j +l and 0 l k +j and (m;l) 6= (j;0),plus a remainder term.
We now x",A and D and consider (t;s;) in the range dened in the statement
of the proposition.In the rest of the proof,when the Landau notion O is used,we
implicitly mean that the corresponding constant depends only on ,",A,D,k and
j.
Let j 0 and k + j 0.Now suppose H
m;k+12l
holds for all 0 m j + l
and 0 l k +j except when (m;l) = (j;0).We are going to show that it implies
H
j;k+1
.Since jj D
log n
n
D
log p
p
,we have for all these values of m and l:
l
E(
p1
U
m
p1
) = O(
log
l+2(k+12l)+3m
p
p
(k+1)=2
) = O(
log
2(k+1)+3j
p
p
(k+1)=2
)
Hence,
E(
p
U
j
p
) = (t;s)E(
p1
U
j
p1
) +O(
log
2(k+1)+3j
p
p
(k+1)=2
)
Therefore,recursively on p,we obtain for all n
E(
n
U
j
n
) =
n
X
p=1
(s;t)
np
O(
log
2(k+1)+3j
p
p
(k+1)=2
)(26)
= n(s;t)
n=2
O(n
jkj
) +(1 +:::+(t;s)
n=2+1
)O(
log
2(k+1)+3j
n
n
(k+1)=2
)
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 19
As above,since is aperiodic,it follows that the law of (X;Y ) on R
2
is aperiodic.
Hence
sup
"<s
2
+t
2
<A
j(t;s)j < 1
therefore (26) yields
E(
n
U
j
n
) = O(
log
2(k+1)+3j
n
n
(k+1)=2
)
Thus we obtain H
j;k+1
.
Now,we note that from Lemma 5.1,H
j;j
holds for all j 0.This also guarantees
H
j;k
when k + j 0.Then we proceed by induction on h = j + k.From the
considerations above,we obtain H
j;k
fromthe knowledge of other H
j
0
;k
0 with j
0
+k
0
<
j +k.So we are done.
Remark 5.3.If we make the following additional assumption on
sup
t
2
+s
2
1
j(t;s)j < 1
then Proposition (5:2) holds uniformly in A and
E(
n
) = O(
1
n
k
)
holds uniformly in (t;s;) when 2 [D
log n
n
;D
log n
n
] and t
2
+s
2
"and the constant
in O depends only";k;D and .
5.3.Estimate for intermediate values of s and t.Now we will treat the case
when jj D
log n
n
and D
log
8
n
p
n
j(s;t)j "for some xed (large) D > 0.
Proposition 5.4.With the notation above,there is some"="() > 0,such that
uniformly in ,jj D
log n
n
Z
D
log
8
n
p
n
j(s;t)j"
jE(
n
(t;s;))jdtds = O(
1
nlog
2
n
)
where the constant in O depends on and D.
Proof.Let us write r:= j(t;s)j =
p
t
2
+s
2
and now choose"="() > 0 so that
for j(t;s)j ",j@
1
(t;s)j = O
(r) and j@
2
(t;s)j = O
(r).Choosing"smaller if
necessary,we can assume that when j(t;s)j "
9C = C() > 0;j(t;s)j e
Cr
2
or equivalently,
1
1 j(t;s)j
= O
(
1
r
2
)
20 E.BREUILLARD
We now suppose that jj D
log n
n
.Until the end of the proof,when we use
Landau's notation O we implicitly mean that the underlying constant depends only
on and D.
Fromthe computation in the subsection above (with j = 2 and k = 1),we obtain
that E(
p
U
2
p
) is a sum of (t;s)E(
p1
U
2
p1
) and a linear combination with bounded
coecients of the terms E(
p1
U
m
p1
) with 0 m 2,E(
p1
),E(
p1
U
p1
)@
2
(t;s),
and E(
p1
U
3
p1
)(@
1
(t;s)+@
2
(t;s)),with a remainder of order O
;D
(log
6
n).Hence,
making use of Lemma 5.1,for all n and p,p n,and if D
log
2
n
p
n
r ",we have
E(
p
U
2
p
) = (t;s)E(
p1
U
2
p1
) +rO(
p
nlog
4
n)
where O is independent of p and n.For any given p,
n
2
p n,we iterate this
equation
n
4
times.We obtain:
E(
p
U
2
p
) =
n=4
(s;t)O(nlog
2
n) +(1 +:::+(t;s)
n=4
)rO(
p
nlog
4
n)
=
1
1 j(t;s)j
rO(
p
nlog
4
n)
=
1
r
O(
p
nlog
4
n)(27)
since
n=4
(s;t)O(nlog
2
n) = e
Cnr
2
=4
O(nlog
2
n) =
1
r
O(
p
nlog
4
n),because D
log n
p
n
r ".
Similarly we can express E(
p
) as above (taking j = 0 and k = 1 in the proof
of prop.5.2) as a sum of (t;s)E(
p1
) and a linear combination with bounded
coecients of the terms E(
p1
) and E(
p1
U
p1
)(@
1
+@
2
) with a rest of order
O(
log
4
n
n
):Since D
log
2
n
p
n
r "we have
E(
p
) = (t;s)E(
p1
) +rO(
log
2
n
p
n
)
Iterating as above,we obtain for
n
2
p n
(28) E(
p
) =
1
r
O(
log
2
n
p
n
)
Nowwe look at E(
p
U
p
).As above (take j = 1,k = 0) it is a sumof (t;s)E(
p1
U
p1
)
and a linear combination with bounded coecients of the terms @
2
(t;s)E(
p1
),
E(
p1
);E(
p1
U
p1
);and (@
1
+@
2
)E(
p1
U
2
p1
);with a rest of order O(
log
5
n
p
n
).
But thanks to (28) and (27) all these terms are O(
log
5
n
p
n
) when
n
2
p n.Hence,
E(
p
U
p
) = (t;s)E(
p1
U
p1
) +O(
log
5
n
p
n
)
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 21
And then,iterating this relation,for all p,
3n
4
p n,we get
(29) E(
p
U
p
) =
1
r
2
O(
log
5
n
p
n
)
Finally we again decompose E(
p
) but pushing one step further the Taylor ex
pansion and we see by the above calculation (take j = 0,k = 2) that it is a sum
of (t;s)E(
p1
) and a linear combination with bounded coecients of the terms
E(
p1
),E(
p1
U
p1
)@
1
(t;s);
2
E(
p1
),
2
E(
p1
U
p1
) and
2
E(
p1
U
2
p1
) with
a rest of order O(
log
6
n
n
p
n
):Thanks to (27),(28),(29) and Lemma 5.1,they are all of
order at most
1
r
O(
log
6
n
n
p
n
) when
3n
4
p n.Thus
E(
p
) = (t;s)E(
p1
) +
1
r
O(
log
6
n
n
p
n
)
and consequently for large n
E(
n
) =
1
r
3
O(
log
6
n
n
p
n
)
Then,integrating over r when D
log
8
n
p
n
r "we obtain:
Z
D
log
8
n
p
n
j(s;t)j"
jE(
n
)jdtds O(
log
6
n
n
p
n
)
Z
D
log
8
n
p
n
r"
1
r
3
rdr
O(
1
nlog
2
n
)
This concludes the proof of the proposition.
6.Study of a dynamical system
In this section we study a quadratic dynamical system in the complex plane and
give precise estimates that will be crucial in the proof of the domination condition
in the next section (i.e.Proposition 7.2).
We rst x three real numbers x;y;z satisfying the following condition
:= det
0@
1 x z
x 1 y
z y 1
1A
= 1 +2xyz x
2
y
2
z
2
0
22 E.BREUILLARD
And suppose additionally that jxj 1,jyj 1 and jzj < 1.Now dene the following
three sequences recursively:
a
k+1
= a
k
+
1
2
2
2
c
2k
+2ic
k
y(30)
b
k+1
= b
k
+
1
2
2
2
b
2k
+2ib
k
z
c
k+1
= c
k
+
x
2
2
2
b
k
c
k
+ib
k
y +ic
k
z
where the initial values a
0
;b
0
and c
0
are arbitrary and is a given real number.In
the applications below, will be small (of order log(n)=n) and the above dynamical
system can be viewed as a perturbation of that given when = 0.In this section we
will study the behavior of the above three sequences depending on initial values and
also on the values of the parameters x;y;z and .
We rst note that (b
k
) is a quadratic dynamical system and is therefore conjugate
to P
:u 7!u
2
+c
for some complex number c
.A straightforward computation
shows that if we set
x
k
:=
1
2
+iz 2
2
b
k
then we have x
k+1
= P
(x
k
) where c
=
1
4
2
(1 z
2
).In the limit when tends to
0,then c
tends to
1
4
,which is on the boundary of the Mandelbrot set.As long as
is small enough and nonzero,then the Fatou set corresponding to P
has exactly one
bounded connected component.Moreover,as soon as jj < 1,for every starting point
x
0
lying inside this component,the resulting sequence of iterates (x
k
) will converge
to the attracting xed point x
given by
x
=
1
2
jj
p
1 z
2
Thus for 6= 0 and jj < 1,the sequence (b
k
) converges to
b
=
iz +sgn()
p
1 z
2
2
Now let us dene
v
k
:= c
k
+
yz x
1 z
2
b
k
Then it is easy to check directly from the equations (30) that v
k
satises
(31) v
k+1
v = (x
k
+
1
2
)(v
k
v)
where(32) v =
i
2
y xz
1 z
2
Let us write y
k
=
1
2
+x
k
,then we obtain
(33) v
k
v = (v
0
v)y
0
::: y
k1
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 23
Similarly,if we let
f
k
:= a
k
+
yz x
1 z
2
c
k
then we nd that
f
k+1
f
k
=
2(1 z
2
)
+(v
k
v)(1 (
1
2
+x
k
))
yz x
1 z
2
2
2
(v
k
v)
2
Making use of (31),it follows that for k 1
a
k
= f
0
+
yz x
1 z
2
2
b
k
+
yz x
1 z
2
(v
0
2v
k
) +(34)
k
2(1 z
2
)
2
2
(v
0
v)
2
k1
X
p=0
(y
0
::: y
p1
)
2
We are now going to study the dynamical system (30) in the particular case when
the initial values are dened by
a
0
= 0(35)
b
0
=
i
2
z
c
0
=
i
2
w
where w is a xed real number.Then x
0
=
1
2
belongs to the lled Julia set of P
,
and the sequence (x
k
) (hence (y
k
) too) stays on the real line and satises
1
2
jj
p
1 z
2
= x
x
k
1
2
Additionally,
v
0
=
i
2
(w +z
yz x
1 z
2
)
Together with (31) and (32) this shows that v
k
belongs to iR for all k.With these
initial values,(34) takes the form
a
k
=
1
2
yz x
1 z
2
2
(iz +
1
(1 y
k
)) +
i
2
yz x
1 z
2
(2w
2y
1 z
2
+z
yz x
1 z
2
)
i
yz x
1 z
2
(w y)y
0
::: y
k1
+
+k
2(1 z
2
)
+
1
2
(w y)
2
k1
X
p=0
(y
0
::: y
p1
)
2
24 E.BREUILLARD
taking the real part we get
Re(a
k
) =
1
2jj
yz x
1 z
2
2
1 y
k
jj
+k
2(1 z
2
)
+(36)
1
2
(w y)
2
k1
X
p=0
(y
0
::: y
p1
)
2
The following lemma summarizes the computations above and encloses the infor
mation that will be relevant to the sequel:
Lemma 6.1.In the dynamical system dened by (30),with initial values given by
(35),the following holds.
Take jj 1=2.Then for all k 0
(i) jkj 1=
p
1 z
2
implies 1 y
k
k
1
2
2
(1 z
2
);and Re(b
k
) k
1z
2
4
;and
Re(a
k
)
k
4
[
(yz x)
2
1 z
2
+2e
4
(w y)
2
]
(ii) jkj 1=
p
1 z
2
implies jj
p
1 z
2
1 y
k
1
2
jj
p
1 z
2
;and Re(b
k
)
1
4jj
p
1 z
2
and
Re(a
k
)
1
jj
1
4
p
1 z
2
[
(yz x)
2
1 z
2
+
1
4
(w y)
2
]
(iii) jb
k
j
1
2
and Re(b
k
) 0
(iv) jv
k
j 2=(1 z
2
) +jwj +1
(v) jc
k
j 3=(1 z
2
) +jwj +1
(vi) Re(a
k
)Re(b
k
) Re(c
k
)
2
(vii) Re(a
k
) is a nondecreasing sequence.
Proof.All these points are easy to check from what was done above.The proof of
(i) follows by induction;it is true when k = 0 and,assuming the inequality for k;
we get
1 y
k+1
= (1 y
k
)y
k
+
2
(1 z
2
)(37)
k
1
2
2
(1 z
2
)(1 jj
p
1 z
2
) +
2
(1 z
2
)
k
1
2
2
(1 z
2
) +
1
2
2
(1 z
2
)(2 jkj
p
1 z
2
)
(k +1)
1
2
2
(1 z
2
)
Besides,for all p k we have y
p
1 jj
p
1 z
2
e
2jj
p
1z
2
since 1 t e
2t
if
t 2 [0;
1
2
].Hence
(y
0
::: y
p1
)
2
e
4jpj
p
1z
2
e
4
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 25
The inequality for Re(a
k
) in (i) now follows instantly from (36) and the fact that
0.
Point (ii) follows from the fact (granted by (37)) that 1 y
k
cjj implies 1
y
k+1
cjj for any real number c with
p
1 z
2
c 0.
We have 2b
k
= iz + (1 y
k
)= and 0 1 y
k
jj
p
1 z
2
so j2b
k
j
p
z
2
+(1 z
2
) 1 yields (iii).Similarly v
k
v = (v
0
v)y
0
::: y
k1
,hence
2v
k
= i(y xz)=(1 z
2
) +i(w y)y
0
::: y
k1
Since jy
i
j 1 for all i,we get j2v
k
j jy xzj=j1 z
2
j + jw yj so j2v
k
j
2=j1z
2
j+jwj+1,and we have (iv) and also (v) because of (iii) and v
k
:= c
k
+
yzx
1z
2
b
k
:
For (vi);we have Re(a
k
)
1y
k
2
2
yzx
1z
2
2
= Re(b
k
)
yzx
1z
2
2
,so (recall that the v
k
belong to iR)
Re(a
k
)Re(b
k
)
Re(b
k
)
yz x
1 z
2
2
= Re(c
k
)
2
Finally (vii) is easily checked from (36).
7.Proof of the domination condition for small values of the
parameters s;t;
In this section,we give a domination estimate for a particular type of trigonometric
sum that will arise in the proof of local limit theorem.We then apply these estimates
to treat the part of the integral that yields a contribution to the limit,that is when
s;t; are small.
7.1.Estimating a trigonometric sum.Let us consider throughout this section a
sequence of independent and identically distributed randomvariables (A
k
;B
k
;C
k
;D
k
)
k1
in R
4
.We assume that the distribution has compact support in R
4
;and that
E(A
k
) = E(B
k
) = E(C
k
) = 0.Let us use the shorthand
X to denote the expectation
E(X) of a random variable X.We x the following notations for the correlations
x =
A
1
B
1
q
A
21
B
2
1
;y =
A
1
C
1
q
A
21
C
2
1
;z =
B
1
C
1
q
B
2
1
C
2
1
We also assume that A
1
is not identically 0 and that the distribution of the marginal
(B
k
;C
k
) is not degenerate,i.e.is not supported on a line.This is equivalent to the
condition jzj < 1.
Fix w 2 Rand then consider the trigonometric product for r; in R,
(38)
n
=
Y
1kn
e
ir(A
k
=
p
A
21
wC
k
=
p
C
2
1
)
e
iD
k
!
Y
1p<qn
e
iB
q
C
p
=
p
B
2
1
C
2
1
!
e
i
2
z(C
1
+:::+C
n
)
2
=
C
2
1
The following proposition yields the desired estimate for the trigonometric sumE(
n
).
26 E.BREUILLARD
Proposition 7.1.Let us x D > 0 a positive number and m 1 an integer.
For any integer n 1 and any distribution (A
1
;B
1
;C
1
;D
1
),of compact support
and as described above,the following estimate holds uniformly when r varies in
[D
log
2m
n
p
n
;D
log
2m
n
p
n
] and varies in [D
log n
n
;D
log n
n
]n[
2
n
p
1z
2
;
2
n
p
1z
2
],
jE(
n
)j exp(
njj
p
1 z
2
8
+
r
2
jj
C
16
p
1 z
2
) +
1 +jwj
1 z
2
4
O(
log
6m
n
p
n
)
where C =
(w y)
2
+
(yzx)
2
1z
2
.
Similarly,if r varies in [D
log
2m
n
p
n
;D
log
2m
n
p
n
] and varies in [
2
n
p
1z
2
;
2
n
p
1z
2
],we
obtain
jE(
n
)j exp[nr
2
Ce
4
4
] +
1 +jwj
1 z
2
4
O(
log
6m
n
p
n
)
The constant in O() depends only on D and on the size of the distribution
M = maxfjA
1
j=
q
A
21
;jB
1
j=
q
B
2
1
;jC
1
j=
q
C
2
1
;jD
1
jg
Proof.Let n 2 N and r; 2 R be as in the statement of the proposition.Let U
0
= 0
and for k 1,
U
k
= (C
1
+:::+C
k
)=
p
C
2
We rst set a few notations.Let q
k
be the quadratic form
q
k
(u;v) = a
k
u
2
+b
k
v
2
+2c
k
uv
where the coecients are dened right below.Then let
0
= 1 and for k 1
k
= e
ir(A
1
+:::+A
k
)=
p
A
21
Y
1p<qk
e
iB
q
C
p
=
p
B
2
1
C
2
1
!
e
i(D
1
+:::+D
k
)
And nally set
P
k
(r;) = E(
nk
e
q
k
(r;U
nk
)
)
We shall dene the coecients a
k
;b
k
;c
k
recursively as follows.We set:
a
k+1
= a
k
+
1
2
2
2
c
2k
+2ic
k
y(39)
b
k+1
= b
k
+
1
2
2
2
b
2k
+2ib
k
z
c
k+1
= c
k
+
x
2
2
2
b
k
c
k
+ib
k
y +ic
k
z
and the initial values are taken to be:a
0
= 0;c
0
=
i
2
w (recall that w 2 R),and
b
0
=
i
2
z.
These relations are precisely the recurrence relations (30) dened in the last section.
As was discussed there,Lemma 6.1 applies and,as soon as jj D
log n
n
and jrj
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 27
D
log
2m
n
p
n
,we see that b
k
and c
k
are uniformly bounded (by const=(1 z
2
) +jwj +
1) and that Re(a
k
)Re(b
k
) (Re(c
k
))
2
.Also note that P
0
= E(
n
e
q
0
(r;U
n
)
) =
E(
n
e
i
2
zU
2
n
riwU
n
),so that
P
0
(r;) = E(
n
):
And
P
n
= E(e
q
n
(r;0)
) = e
a
n
r
2
:
In order to obtain the desired estimate on E(
n
),that is on P
0
(r;),we are going
to start from P
0
(r;) and proceed by induction on k.To achieve this goal,we need
the following key induction step:
Claim:For all k and n with k n 1,and if jj D
log n
n
and jrj D
log
2m
n
p
n
,we
have
(40) P
k
(t;s) = e
b
k
2
+i
D
P
k+1
(t;s) +
1 +jwj
1 z
2
4
O(
log
6m
n
n
p
n
)
where the constant in O depends only on D and on the size M of the support of the
distribution (A
1
;:::;D
1
).
This crucial estimate follows from the computation made below.First,note that
q
k
(r;U
nk
) = q
k
(r;U
nk1
+C
nk
=
p
C
2
)
= q
k
(r;U
nk1
) +b
k
2
C
2
nk
=
C
2
+
2b
k
2
U
nk1
C
nk
=
p
C
2
+2c
k
rC
nk
=
p
C
2
We also recall that A
n
is the event fmax
0kn
jU
k
j
p
nlog ng.
We will use the expansion e
t
= 1 +t +t
2
=2 +O(t
3
) near t = 0.Applying Lemma
5.1 and bearing in mind that is centered we can write
P
k
= E
nk1
e
irA
nk
=
p
A
2
e
iB
nk
U
nk1
=
p
B
2
e
iD
nk
e
q
k
(r;U
nk
)
= E(1
A
n
nk1
e
q
k
(r;U
nk1
)
e
irA
nk
=
p
A
2
e
iB
nk
U
nk1
=
p
B
2
e
iD
nk
e
b
k
2
C
2
nk
=
C
2
2b
k
2
U
nk1
C
nk
=
p
C
2
2c
k
rC
nk
=
p
C
2
) +e
c
0
log
2
n
= E(1
A
n
nk1
e
q
k
(r;U
nk1
)
(1 +irA
nk
=
p
A
2
+iB
nk
U
nk1
=
p
B
2
+
iD
nk
b
k
2
C
2
nk
=
C
2
2b
k
2
U
nk1
C
nk
=
p
C
2
2c
k
rC
nk
=
p
C
2
+
1
2
"
irA
nk
=
p
A
2
+iB
nk
U
nk1
=
p
B
2
+iD
nk
b
k
2
C
2
nk
=
C
2
2b
k
2
U
nk1
C
nk
=
p
C
2
2c
k
rC
nk
=
p
C
2
#
2
+O(
log
6m
n
n
p
n
))) +e
c
0
log
2
n
28 E.BREUILLARD
Expanding further,we get
P
k
= E(
nk1
e
q
k
(r;U
nk1
)
[1 b
k
2
+i
D r
2
(
1
2
2
2
c
2k
+2ic
k
y)
2
U
2
nk1
(
1
2
2
2
b
2k
+2ib
k
z) 2rU
nk1
(
x
2
2
2
b
k
c
k
+ib
k
y +ic
k
z)]
+O(
log
6m
n
n
p
n
))
= E(
nk1
e
q
k
(r;U
nk1
)
[exp
b
k
2
+i
D a
k+1
r
2
b
k+1
2
U
2
nk1
2c
k+1
rU
nk1
+O(
log
6m
n
n
p
n
)] +O(
log
6m
n
n
p
n
)
where the coecients a
k+1
,b
k+1
and c
k+1
are the ones dened above.Hence we
indeed obtain
P
k
= e
b
k
2
+i
D
P
k+1
+O(
log
6m
n
n
p
n
)
This computation makes sense as long as b
k
and c
k
remain uniformly bounded
when n grows,and we checked that it is indeed the case.We also need to insure that
Re(q
k
) 0 everywhere,i.e.Re(a
k
)Re(b
k
) (Re(c
k
))
2
and Re(a
k
) 0,but we also
checked that above.
With the help of Lemma 5.1 and the remark following it,we verify that the constant
involved in the O in the above calculations can be taken of the form c D
4
K
4
;where
c > 0 depends only on the size of the support of the distribution of (A
1
;B
1
;C
1
;D
1
)
and where K is that number that bounds c
k
and b
k
.Hence the claim (40) is
proved.
We can now iterate (40),going from P
0
to P
n
.We deduce
E(
n
) = e
2
P
n1
k=0
b
k
e
in
D
e
a
n
r
2
+O(
log
6m
n
p
n
)
The constant involved here in the O is bounded by some cD
4
(1 +jwj)
4
=(1 z
2
)
4
where c is a constant depending only on M = maxfjA
1
j=
q
A
21
;jB
1
j=
q
B
2
1
;jC
1
j=
q
C
2
1
;jD
1
jg.
jE(
n
)j e
2
P
n1
k=0
Reb
k
e
r
2
Rea
n
+O(
log
6m
n
p
n
)
From Lemma 6.1,if jj
2
p
1z
2
1
n
then Reb
k
1
4jj
p
1 z
2
for all k n=2 and
Re(b
k
) 0 for all k.So the rst factor in the above equation leads to the bound
e
njj
p
1z
2
=8
.While Re(a
n
)
1
jj
1
4(1z
2
)
1=2
1
4
(w y)
2
+
(yzx)
2
1z
2
.
If jj
2
p
1z
2
1
n
,then Re(a
n
) Re(a
n=2
)
n
4
e
4
(w y)
2
+
(yzx)
2
1z
2
.
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 29
So we obtain the desired inequalities.
7.2.Domination condition.The purpose of this subsection is to give the precise
estimate we wanted in the course of the proof of the local limit theorem (control of
the part of the integral where all parameters s;t; are small).This is explained is
the following proposition:
Proposition 7.2.Let be a probability measure on the Heisenberg group with the
properties described in the introduction.Let
n
be the random variable dened in
(19) at the beginning of section 5.There exist positive numbers c
1
> 0 and c
2
> 0
depending only on ,such that,if D > 0 denotes some positive number,then the
following estimates hold uniformly
(i) when s and t vary in [D
log
8
n
p
n
;D
log
8
n
p
n
] and varies in [D
log n
n
;D
log n
n
]n[
c
2
n
;
c
2
n
],
jE(
n
)j exp(c
1
njj +
t
2
+s
2
jj
) +O(
log
24
n
p
n
)
(ii) when s and t vary in [D
log
8
n
p
n
;D
log
8
n
p
n
] and varies in [
c
2
n
;
c
2
n
],
jE(
n
)j exp(c
1
n(t
2
+s
2
)) +O(
log
24
n
p
n
)
where the constant in O depends only on and on D.
Proof.We are going to apply the results of the last section.In order to do so,we
need to choose carefully the variables A
1
;B
1
;C
1
;D
1
as well as the coecient w.Take
D
1
= Z=
p
X
2
Y
2
,C
1
= Y=
p
Y
2
,B
1
= X=
p
X
2
and A
1
= cos()
X
p
X
2
+2 sin()
Y
p
Y
2
and w =
sin()
()
,where ()
2
=
A
21
= 1 +3 sin
2
() +4z sin() cos() and () 0.An
easy computation shows that 8 ()
2
(1 z
2
)=5.Clearly the variables A
1
;B
1
and C
1
are linearly dependent,hence = 0.But B
1
and C
1
are linearly independent
since is aperiodic,hence jzj < 1.
Then the two expressions
n
in (19) and (38) agree if we change into =
p
X
2
Y
2
and let r and be determined by
t =
r cos()
()
p
X
2
s =
r sin()
()
p
Y
2
30 E.BREUILLARD
From the above inequality on (),we conclude that if M
X;Y
= maxf
1
X
2
;
1
Y
2
g and
m
X;Y
= minf
1
X
2
;
1
Y
2
g;
m
X;Y
1 z
2
5
(t
2
+s
2
)
r
2
X
2
Y
2
8M
X;Y
(t
2
+s
2
)
To compute the constant C appearing in Proposition (7.1),let us rst compute x;y
and z.
z =
XY =
p
X
2
Y
2
y =
z cos() +2 sin()
()
x =
cos() +2z sin()
()
hence
C = (w y)
2
+
(yz x)
2
1 z
2
=
sin() +z cos()
()
2
+
(1 z
2
) cos
2
()
()
2
=
1
()
2
[1 2z cos() sin()]
1
()
2
(1 jzj)
1 z
2
2 8
Also note that jwj 1=()
p
5=(1 z
2
).The rst estimate in (7:1) now yields
jE(
n
)j exp(
njj
p
X
2
Y
2
p
1 z
2
8
+
r
2
jj
C
4
p
1 z
2
) +O(
log
6
n
p
n
)
exp(
p
X
2
Y
2
p
1 z
2
njj
8
+
r
2
jj
X
2
Y
2
1
64
) +O(
log
6
n
p
n
)
exp(
q
X
2
Y
2
XY
2
njj
8
+
M
X;Y
(t
2
+s
2
)
64 jj
) +O(
log
6
n
p
n
)
exp(c
1
njj +
(t
2
+s
2
)
jj
) +O(
log
6
n
p
n
)
where we can take c
1
p
X
2
Y
2
XY
2
8
minf1;M
X;Y
=8g,and the constant in O in
the last line depends only on and D.
We thus have obtained (i),and (ii) follows similarly with c
1
e
4
80
(
X
2
Y
2
XY
2
).
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 31
8.Proofs of the main theorems
We can now nish the proof of Theorem 2.1.Let f and g be as in the statement
of Theorem 2.1.Recall that the number was dened in (5).As was remarked at
the end of section 3,we can write
n
2
h(
n
)f;gi = A
n
+B
n
where A
n
and B
n
are dened as follows:
A
n
= n
2
Z
jjD
log n
n
h
(
n
)f
;g
i
H
d
and there is a constant C depending on D, and the size of the set f;9(t;s);F
g
(g)(s;t;) 6=
0g such that
(41) jA
n
j
C
n
k
0
2
kfk
L
2
(G)
kgk
L
2
(G)
as soon as the integer k
0
satises D > k
0
=c (where c > 0 is a constant depending on
only dened in (13)).And
(42) B
n
=
Z
G
dx
(2)
3=2
f(x)I
n
(x)
with
I
n
(x) = n
2
Z
jjD
log n
n
E(
n
(t;s;));e
i
F
(g)(t y;s +2y;)
L
2
(R
2
)
d
where,keeping the notations of Theorem 2.1 F
(g) is the Fourier transform dened
in (4),
n
(t;s;) is dened in (19) and x = (x;y;z) 2 G.
Splitting the integral on R
2
in the expression of I
n
(x) above into the parts when
j(t;s)j D
log
8
n
p
n
on the one hand and j(t;s)j D
log
8
n
p
n
on the other hand,we can
write:(43) I
n
(x) = I
S
n
(x) +I
L
n
(x)
and assert that if kxk = maxfjxj;jyj;jzjg and 2jj are less than say K > 1 then
F
(g)(t y;s + 2y;) 6= 0 implies that maxfjtj;jsj;jjg A(1 + K
2
) where
A > 0 is a number such that the support of F
(g) lies inside [A;A]
3
.Then we may
write:
jI
L
n
(x)j n
2
kF
(g)k
1
Z
jjD
log n
n
Z
D
log
8
n
p
n
j(t;s)jA(1+K
2
)
jE(
n
(t;s;))jdtdsd
n
2
kF
(g)k
1
Z
jjD
log n
n
O(
1
nlog
2
(n)
)d(44)
kF
(g)k
1
O(
1
log(n)
)
32 E.BREUILLARD
where line 44 is granted by the two Propositions 5.2 and 5.4.The constant involved
here in O depends only on ,D and the size A of the support of F
(g) and on the
maximum of kxk.In particular it is uniform when x varies in compact subsets of G.
We can now concentrate on the part I
S
n
(x) of the integral which actually gives a
contribution to the limit.From Proposition 7.2,we deduce that if j(t;s)j Dlog
8
n
and c
2
jj Dlog n
jE(
n
(
t
p
n
;
s
p
n
;
n
))j e
c
1
(jj+t
2
=jj+s
2
=jj)
+O(
log
24
n
p
n
)
and if j(t;s)j Dlog
8
n and jj c
2
jE(
n
(
t
p
n
;
s
p
n
;
n
))j e
c
1
(t
2
+s
2
)
+O(
log
24
n
p
n
)
where the constant in O depends only on and D.But one can check that the
function
(t;s;) 7!e
c(jj+t
2
=jj+s
2
=jj)
is integrable over R
3
.And
(t;s;) 7!e
c(t
2
+s
2
)
is integrable in (t;s;) 2 R
2
[c
2
;c
2
].Moreover
Z
jjDlog n
Z
j(t;s)jDlog
8
n
O(
log
24
n
p
n
) = O(
log
41
n
p
n
)!0
And nally,from the central limit theorem (see [Tut] or [Rau]) the following limit
holds pointwise in t;s;:
E(
n
(
t
p
n
;
s
p
n
;
n
))!E(e
i(tX+sY +(ZY
2
))
)
where (X;Y;Z) is the limit random variable with gaussian distribution
1
as men
tioned in the introduction.Hence if we use the shorthand t
n
:= t=
p
n,s
n
= s=
p
n and
n
:= =n we have (see the denition of
n
= (
t
p
n
;
s
p
n
;
n
) in (18) above),uniformly
when x varies in compact subsets of G
lim
n!+1
E(
n
(t
n
;s
n
;
n
))e
i
n
F
(g)(t
n
n
y;s
n
+2
n
y;
n
) = E(e
i(tX+sY +(ZY
2
))
)
F
(g)(0)
Since the heat kernel corresponding to is a fastly decreasing smooth function
p(x;y;z) on R
3
(see [VSC]) it follows that E(e
i(tX+sY +(ZY
2
))
) is integrable in
(t;s;) 2 R
3
and we compute by the Fourier inversion formula
Z
R
3
E(e
i(tX+sY +(ZY
2
))
)dtdsd = (2)
3
p(0;0;0) = (2)
3
c()
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 33
Therefore,by Lebesgue's dominated convergence theorem,we get uniformly when
x varies in compact subsets
lim
n!+1
I
n
(x) = lim
n!+1
I
S
n
(x)
= n
2
Z
jjD
log n
n
Z
j(t;s)jD
log
8
n
p
n
E(
n
(t;s;))e
i
F
(g)(t y;s +2y;)dtdsd
= (2)
3
c()
F
(g)(0)
Integrating in x we nally obtain
lim
n!1
B
n
= lim
n!+1
Z
G
dx
(2)
3=2
f(x)I
n
(x)
= (2)
3=2
c()
F
(g)(0)
Z
G
f
= c()
Z
G
f
Z
G
g(45)
Combining (45) and (41) we get
(46) lim
n!1
n
2
h(
n
)f;gi = c()
Z
G
f
Z
G
g
as desired.
We also remark that any translation of g (to the left
z
g(x) = g(z
1
x) or to the
right g
z
(x) = g(xz)) has again a compactly supported Fourier transform F
(
z
g) and
F
(g
z
):If z remains in a compact subset,then the supports of F
(
z
g) and F
(g
z
)
also remain within a prescribed bounded set.Moreover kF
(g
z
)k
1
= kF
(
z
g)k
1
and
kg
z
k
2
= k
z
gk
2
are independent of z.This follows from the computation of F
(
z
g)
which yields
(47) F
(
z
g)(t;s;) = e
i
F
(g)(t;s (2z
y
z
x
);)
where z = (z
x
;z
y
;z
z
) and = z
x
t+z
y
s+(z
z
z
2
y
).Consequently all the calculations
above,and (46) in particular,hold uniformly for translates of g on compact subsets.
Now take f
n
a Dirac sequence of positive functions supported on a neighborhood
of 0 of diameter of order 1=n
3
.Then kf
n
k
22
= O(n
9
).Choosing D large enough (so
that A
n
in (41) remains negligeable) we see that we can replace f by f
n
in the above
calculations,hence
(48) lim
n!1
n
2
h(
n
)f
n
;gi = c()
Z
G
g
And the same holds uniformly when z varies in compact subsets and g is replaced
by g
z
or
z
g.
34 E.BREUILLARD
Assume now that F
(g) is also absolutely continuous (that is belongs to W
1;1
(R
3
),
i.e.its derivative distribution is a function in L
1
(R
3
)),then it follows
1
that g
is
real analytic and that kxk d
x
(g
) is a bounded function on R
3
(where kxk is the
max of the coordinates of x 2 R
3
).For any compact K we can then nd a constant
C = C(g;K) such that whenever y is small enough
supz2K
k
y
1
z
g
z
gk
1
C kyk
and
sup
z2G
k
y
1g
z
g
z
k
1
Ckyk
Hence uniformly for z in compact subsets,k
z
g f
n
z
gk
1
= O(1=n
3
) and kg
z
f
n
g
z
k
1
=
O(1=n
3
).From (48) it now follows that uniformly for z in compact subsets:
lim
n!1
n
2
Z
g(z
1
x)d
n
(x) = c()
Z
G
g
and(49) lim
n!1
n
2
Z
g(xz)d
n
(x) = c()
Z
G
g
We now deduce Theorem 1.1 from Theorem 2.1.
Proof.For all"> 0 one can nd
2
a strictly positive function h in L
1
(R
3
) such
that
b
h is C
1
and compactly supported,and such that there exists C > 0,with
h(x) kxk
6+"
C for x 2 R
3
large enough.Let g = h
1
.Now for all 2 R
3
and
z 2 G,e
i
1
(x)
g
z
(x) also has C
1
Fourier transform F
of compact support,hence
(49) holds for it uniformly when z varies in compact subsets.By Levy's criterion for
weak convergence of nite measures,this shows that the sequence of nite measures
d
z
n
= n
2
g
z
d
n
converges weakly (in the space of nite measures on G) to c()g
z
(x)dx
uniformly when z varies in compact subsets.Now let f be a function on G as in the
statement of the theorem.Then f=g is a bounded continuous function,hence
n
2
Z
f
z
d
n
=
Z
f
z
=g
z
d
z
n
!c()
Z
G
f
uniformly when z varies in compact subsets.
1
Note that if a function h(x) 2 L
1
(R) is such that
b
h(t) 2 C
c
(R) and
d
b
h
dt
2 L
1
(R) (i.e.
b
h absolutely
continuous) then x
dh
dx
is the Fourier transform of
d
dt
(t
b
h),hence is bounded.
2
It is enough to nd a function f 2 L
1
(R) such that
b
f is C
1
of compact support and f(x)
C
1+jxj
2+"
for some C > 0 (e.g.see [Breu] section 3.2).Then take h(x) = f(x)f(y)f(z) if x = (x;y;z):
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 35
9.Uniform local limit theorem for translates of a bounded set
We intend here to prove Theorem 1.2.The probability measure is assumed to
be aperiodic,centered and compactly supported.The letter denotes the associated
gaussian probability distribution.Its density function,the heat kernel,is denoted by
p
1
(x;y;z).It is a fastly decreasing smooth function on G.
We start by xing a nonnegative function K on G such that F
(K) is smooth
and has compact support and F
(K)(0) = 1.Then we form a Dirac family (K
a
)
a>0
by letting K
a
(x) = a
4
K(d
a
(x)).Then F
(K
a
)() = F
(K)(d
1
a
()).Let us also write
K
_
a
(z) = K
a
(z
1
).They also form a Dirac family when a!+1.
Lemma 9.1.There are two sequences of positive numbers ("
n
)
n
and (a
n
)
n
,depending
only on ;with"
n
!0 and a
n
!+1,such that for all bounded Borel sets B G
for which maxfjy
1
j;jy
2
jg n= log n whenever y = (y
1
;y
2
;y
3
) 2 B,and all x 2 G,if
we set P
B
n
(x) =
n
(xB) and Q
Bn
(x) =
n
(xB),the following inequality holds for all
positive integers n;
(50) n
2
P
B
n
K
_
a
n
(x) Q
Bn
K
_
a
n
(x)
"
n
maxf1;jBjg
where jBj denotes the Haar measure of B.
Proof.We may write
P
B
n
K
_
a
(x) =
Z
G
n
(xz
1
B)K
_
a
(z)dz
=
Z
G
E(1
S
1
n
z2B
1
)K
a
(x
1
z)dz
= h(
n
)f;
x
gi
where
x
g is the translate
x
g(z):= g(x
1
z) and f:= 1
B
1 and g(z):= K
a
(z) (note
that K
a
is real).Then we can make use of the calculations performed in the previous
sections to estimate this scalar product.We use the notations introduced in section
8.
As noted at the beginning of section 8,we can write
n
2
h(S
n
)f;
x
gi = A
n
() +B
n
()
where A
n
is controlled by the estimation (41) and B
n
given by (42).Since kfk
L
2
(G)
=
p
jBj and k
x
gk
L
2
(G)
= a
2
kKk
L
2
(G)
we obtain (take k
0
= 3 and D = 3=c)
(51) jA
n
()j C(a)
a
2
n
p
jBj kKk
L
2
(G)
where C(a) is a positive constant depending only on a.Integrating the decomposition
I
n
(y) = I
S
n
(y)+I
L
n
(y) with respect to f(y)dy (see equations (42) and (43));we obtain
that B
n
() can be written as a sum B
S
n
() + B
L
n
().Note that kF
(
x
K
a
)k
1
=
36 E.BREUILLARD
kF
(K)k
1
as follows from (47).Spliting B
L
n
() into part when" j(t;s)j D
log
8
n
n
and the part when j(t;s)j "we have from (42) and (44)
B
L
n
() = B
L
0
n
() +B
L
1
n
()
(52) jB
L
0
n
()j jBj kF
(K)k
1
O
(
1
log n
)
as follows from Proposition 5.4,and
(53) jB
L
1
n
()j
jBj kF
(K)k
1
(2)
3=2
L n
2
sup
Tj(t;s)j";jjD
log n
n
jE(
n
(t;s;))j
where T is the size of the support of the functions F
(
x
g) and L is the Lebesgue
measure of that support:
(t;s) 7!F
(K)(
t y
2
a
;
s +2(y
2
x
2
) +x
1
a
;
a
2
)
where x = (x
1
;x
2
;x
3
) and y = (y
1
;y
2
;y
3
).Note that L is a xed multiple of a
2
.
Since jj D
log n
n
in (44) and by assumption maxfjy
1
j;jy
2
jg n= log n,if we sup
pose additionally that maxfjx
1
j;jx
2
jg 2n= log n then the constant T in the above
equation (53) is bounded by some xed function of a.In Proposition 5.2 we showed
that there is a constant C
T
such that
sup
Tj(t;s)j";jjD
log n
n
jE(
n
(t;s;))j
C
T
n
3
Hence for some number (a) < +1
(54) jB
L
n
()j jBj
(a)
log n
The estimations above can be carried out in a similar way for instead of :In
particular
Q
Bn
K
_
a
n
(x) = h(
n
)f;
x
gi
= A
n
() +B
n
()
The term A
n
() is dealt with in exaclty the same way since estimate (41) is also valid
for and we have
(55) jA
n
()j C(a)
a
2
n
p
jBj kKk
L
2
(G)
In order to control B
n
() we made use of the compact support assumption on .For
we can use the following direct argument because F
(p
1
) is integrable since p
1
(x;y;z)
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 37
decays rapidly when (x;y;z) is large.Recall that (
t
)
t
is a stable semigroup,i.e.
n
=
n
1
= d
p
n
(
1
).From (44) we have
jB
L
n
()j
1
(2)
3=2
jBj kF
(K)k
1
n
2
Z
j(t;s)jD
log
8
n
p
n
jE(
n
(t;s;))jdtdsd
1
(2)
3=2
jBj kF
(K)k
1
Z
j(t;s)jDlog
8
n
jF
(p
1
)(t;s;)jdtdsd
1
(2)
3=2
jBj kF
(K)k
1
o(1)(56)
Therefore it remains to treat B
S
n
() B
S
n
():From (42) and (43),we have
(57)
B
S
n
() B
S
n
()
A n
2
Z
jjD
log n
n
Z
j(t;s)jD
log
8
n
p
n
jE(
n
(t;s;)) E(
n
(t;s;))jdtdsd
with
A =
1
(2)
3=2
jBj kF
(K)k
1
Now the integral on the right hand side tends to 0 as it follows from Lebesgue's
dominated converge theorem like we did in section 8.In section 7 we showed that
there exists an integrable function (t;s;) such that for n suciently large and for
all (t;s;) such that jj Dlog n and j(t;s)j Dlog
8
n,
E(
n
(
t
p
n
;
s
p
n
;
n
))
(t;s;) +O(
log
6
n
p
n
)
On the other hand F
(p
1
) is integrable and
E(
n
(
t
p
n
;
s
p
n
;
n
)) = F
(p
1
)(t;s;)
And by the central limit theorem,we had pointwise
lim
n!1
E(
n
(
t
p
n
;
s
p
n
;
n
)) E(
n
(
t
p
n
;
s
p
n
;
n
)) = 0
Hence by Lebesgue's dominated convergence theorem,
lim
n!+1
n
2
Z
jjD
log n
n
Z
j(t;s)jD
log
8
n
p
n
jE(
n
(t;s;)) E(
n
(t;s;))jdtdsd = 0
Therefore (57) reads:
(58)
B
S
n
() B
S
n
()
jBj o(1)
38 E.BREUILLARD
And nally combining (51),(55),(56),(54) and (58),we have that for some sequence
"
n
!0
n
2
P
B
n
K
_
a
(x) Q
Bn
K
_
a
(x)
"
n
jBj +
(a)
log n
jBj +C(a)
a
2
n
2 kKk
2
p
jBj
whenever x satises maxfjx
1
j;jx
2
jg 2n= log n.But we can choose a sequence
a
n
!+1 such that
(a
n
)
log n
and C(a
n
)
a
2n
n
tend to 0:Changing"
n
if necessary,we
obtain for these values of x
n
2
P
B
n
K
_
a
n
(x) Q
Bn
K
_
a
n
(x)
"
n
maxfjBj;
p
jBjg
Now let us examine the case when x takes large values,i.e.when maxfjx
1
j;jx
2
jg
2n= log n.Then
P
B
n
K
_
a
n
(x) =
Z
G
n
(xzB)K
a
n
(z)dz P(jT
n
j or jU
n
j n=2 log n)+
Z
jzj
n
2 log n
K
a
n
(z)dz
where U
n
is as before the sum Y
1
+:::+Y
n
and T
n
= X
1
+:::+X
n
.Now since F
(K)
is smooth,K decays rapidly and in particular
Z
jzj
n
2 log n
K
a
n
(z)dz =
Z
jzja
n
n
2 log n
K(z)dz = O(
1
n
k
)
for any k 0.Hence we get from Lemma 5.1
P
B
n
K
_
a
n
(x) = o(1=n
2
)
Similarly the same holds for Q
Bn
K
_
a
n
(x) when maxfjx
1
j;jx
2
jg 2n= log n.Changing
("
n
)
n
if necessary,we obtain the desired conclusion,i.e.inequality (50) for all x.
The restrictions on the size of B in the above lemma disappear if we make the
additional assumption (Cramer's condition) that
sup
t
2
+s
2
1
jE(e
i(tX+sY )
)j < 1
Indeed in this case,we can control E(
n
(t;s;)) uniformly for arbitrary large values
of t and s (see remark (5:3)).Therefore we can take T = +1in the estimation (53)
above and the restriction on B is unecessary.We obtain
Lemma 9.2.If we suppose additionally that satises Cramer's condition (9),then
there are two sequences of positive numbers ("
n
)
n
and (a
n
)
n
,depending only on ;
with"
n
!0 and a
n
!+1,such that for all bounded Borel sets B G and all
x 2 G,if we set P
B
n
(x) =
n
(xB) and Q
Bn
(x) =
n
(xB),the following inequality
holds for all positive integers n;
n
2
P
B
n
K
_
an
(x) Q
Bn
K
_
an
(x)
"
n
maxf1;jBjg
where jBj denotes the Haar measure of B.
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 39
Lemma 9.3.Let (a
n
)
n
be a sequence of positive numbers such that a
n
!+1.Then
there exists another sequence ("
n
)
n
with"
n
!0 and a sequence of neighborhoods of
identity (U
n
) converging to identity,such that for all bounded Borel sets B G,the
following inequality holds for all positive integers n;
n
2
Q
Bn
(x) Q
Bn
K
_
a
n
(x)
p(jV
n
BnBj +"
n
jBj)
where jBj denotes the Haar measure of B and where we have set Q
Bn
(x) =
n
(xB)
and p = max
jp
1
(g)j;g 2 G
> 0:
Proof.The proof is straightforward.We rst note that for any bounded Borel set B,
n
(B) pjBj=n
2
.Then we simply write:
jQ
Bn
K
_
a
n
(x) Q
Bn
(x)j
Z
jQ
Bn
(xz) Q
Bn
(x)jK
a
n
(z)dz
Z
z2U
1
n
n
(x(zBB)K
a
n
(z)dz
+2p
jBj
n
2
Z
z =2U
1
n
K
a
n
(z)dz
p(jU
n
BnBj +"
n
jBj)=n
2
where U
n
is a sequence of neighborhoods of identity tending to identity such that
R
z =2U
1
n
K
a
n
(z)dz tends to 0 at innity.
Now,let us complete the proof of Theorem (1:2).Let B be an arbitrary bounded
Borel set satisfying the condition of Lemma (9:1),that is maxfjy
1
j;jy
2
jg n= log n
whenever y = (y
1
;y
2
;y
3
) 2 B (resp.satisfying no additional condition if we assume
Cramer's condition).In the sequel like above,the Landau notations o and O will
correspond to functions depending only on .We keep notations of Lemma (9:1),
P
U
n
B
n
K
_
a
n
(x) =
Z
n
(xzU
n
B)K
a
n
(z)dz
n
(xB)
Z
U
n
K
a
n
(z)dz
where U
n
is as in Lemma 9.3.Now making use of Lemma 9:1 we get uniformly in
x 2 G,
P
B
n
(x) (1 +o(1))P
U
n
B
n
K
_
a
n
(x)(59)
(1 +o(1))[Q
U
n
B
n
K
_
a
n
(x) +(1 +jU
n
Bj)o(1=n
2
)]
40 E.BREUILLARD
And by Lemma 9:3,
Q
U
n
B
n
K
_
a
n
(x) Q
U
n
B
n
(x) +
p
n
2
(jU
2
n
BnBj +"
n
jU
n
Bj)
Q
Bn
(x) +
p
n
2
jU
n
BnBj +
p
n
2
(jU
2
n
BnBj +"
n
jU
n
Bj)
Q
Bn
(x) +
p
n
2
(2jU
2
n
BnBj +"
n
jU
n
Bj)
In particular we have
Q
U
n
B
n
K
_
a
n
(x)
4p
n
2
jU
2
n
Bj
and,from (59);
(60) P
B
n
(x) jU
2
n
BjO(
1
n
2
) +o(
1
n
2
)
Additionally,
(61) P
B
n
(x) Q
Bn
(x) +
2p
n
2
jU
2
n
BnBj +(jU
2
n
Bj +1)o(
1
n
2
)
Now let us turn to the other direction of the inequality.We have,making use of
(60)
P
B
n
K
_
a
n
(x) =
Z
n
(xzB)K
a
n
(z)dz
Z
U
n
n
(xzB)K
a
n
(z)dz +
Z
U
c
n
n
(xzB)K
a
n
(z)dz
n
(xU
n
B)
Z
U
n
K
a
n
(z)dz +(jU
2
n
Bj +1)o(
1
n
2
)
P
U
n
B
n
(x) +(jU
2
n
Bj +1)o(
1
n
2
)
But from (60),
P
U
n
B
n
(x) P
B
n
(x) =
n
(x(U
n
BnB))
jU
2
n
(U
n
BnB)jO(
1
n
2
) +o(
1
n
2
)
Hence
P
B
n
K
_
a
n
(x) P
B
n
(x) +jU
2
n
(U
n
BnB)jO(
1
n
2
) +(jU
2
n
Bj +1)o(
1
n
2
)
Now it follows from Lemma (9:1) that
Q
Bn
K
_
a
n
(x) P
B
n
K
_
a
n
(x) +(1 +jU
n
Bj)o(1=n
2
)
and from Lemma (9:3)
Q
Bn
(x) Q
Bn
K
_
a
n
(x) +
p
n
2
(jU
n
BnBj +"
n
jBj)
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 41
Combining the last three inequalities,we get
(62) Q
Bn
(x) P
B
n
(x) +jU
2
n
(U
n
BnB)jO(
1
n
2
) +(jU
2
n
Bj +1)o(
1
n
2
)
Equations (61) and (62) yield the desired result
(63) jP
B
n
(x) Q
Bn
(x)j jU
2
n
(U
n
BnB)jO(
1
n
2
) +(jU
2
n
Bj +1)o(
1
n
2
)
But clearly
T
n0
U
2
n
(U
n
BnB) is contained in
Bn
B.Hence for every bounded mea
surable set B such that j@Bj = 0 we have
(64) lim
n!0
sup
x2G
n
2
j
n
(xB)
n
(xB)j = 0
And if satises Cramer's condition the estimate (63) holds without the above re
striction on B:Hence (63) holds uniformly on y for all By:We conclude
(65) lim
n!0
sup
x;y2G
n
2
j
n
(xBy)
n
(xBy)j = 0
Remark 9.4.It is easy to see from (63) that the limits in (64) and (65) are uniform
in B when B ranges over the set of balls for a given norm on G lying in a given
compact subset of G.
Finally note that Theorem 1.3 follows instantly from the inequality (60) above.
10.Applications
10.1.An equidistribution result for bounded uniformly continuous func
tions.In this section,we intend to give a proof of Corollary (1:4).The proof splits
into two steps.
Lemma 10.1.Suppose f is a continuous and bounded function on G satisfying
condition (2) of Corollary (1:4) then
lim
n!1
Z
G
f(g)d
n
(g) =`
where =
1
is an arbitrary gaussian measure on G.
Proof.Let (
t
)
t
be the oneparameter semigroup of gaussian measures in which
is embedded.By scaling invariance,
t
coincide with the image of
1
under the
automorphism d
p
t
of G,where d
t
(x;y;z) = (tx;ty;t
2
z).Hence
Z
G
f(g)d
n
(g) =
Z
G
f d
p
n
(g)p(g)dg
where p(g) is the density of .It is known that p as well as its derivatives are smooth,
fastly decreasing functions on G.Let L > 0 be a Lipschitz constant for p in the sense
42 E.BREUILLARD
that jp(g
1
) p(g
2
)j Lkg
1
g
2
k for all g
1
;g
2
2 G where kgk = maxfjxj;jyj;jzjg
C for g = (x;y;z):Fix"> 0 and let C > 0 be such that
Z
kgkC
p(g)dg "
We now have
Z
kgkC
f(d
p
n
(g))p(g)dg
"kfk
1
Let us denote by R(g;h) the rectangle [x;x + h) [y;y + h) [z;z + h) where
g = (x;y;z) 2 G and h > 0.We can nd a subdivision of the hypercube fkgk Cg
by small cubes of the form R(g
i
;h).Hence
Z
kgkC
f(d
p
n
(g))p(g)dg =
X
Z
R(g
i
;h)
f(
p
nx;
p
ny;nz)p(g)dg
Also
X
Z
R(g
i
;h)
f(
p
nx;
p
ny;nz)p(g)dg
X
p(g
i
)
Z
R(g
i
;h)
f(
p
nx;
p
ny;nz)dg
Lhkfk
1
(2C)
3
Now,note that,by viewing R(g;h) as a dierence of several rectangles in R
3
,the
assumption (2) made on f easily implies that for all g 2 G and h > 0
(66) lim
T!+1
1
T
2
Z
f(x)
R(g;h)
(d
1=
p
T
(x))dx =` h
3
since
R(g;h)
d
1=
p
T
can be written as a nite sumof terms of the form
[0;T
1
]:::[0;T
3
]
for some positive or negative T
1;
:::;T
3
.Hence by (66)
lim
T!+1
X
p(g
i
)
Z
R(g
i
;h)
f(
p
nx;
p
ny;nz)dg =`
X
p(g
i
)h
3
But
X
p(g
i
)h
3
1
Z
kgkC
p(g)dg +
X
Z
R(g
i
;h)
jp(g) p(g
i
)j dg
"+Lh(2C)
3
Therefore,combining the above inequalities,for n large enough
Z
f(d
p
n
(g))p(g)dg `
`("+Lh(2C)
3
) +Lhkfk
1
(2C)
3
+"kfk +"
We nally obtain the desired result since h can be taken arbitrarily small.
The second step is about comparing the integrals with respect to the probability
measure and its associated gaussian distribution .Here,we make use of the
uniform version of the local limit theorem (Theorem 1.2).Namely,
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 43
Lemma 10.2.Let f be a bounded and uniformly continuous function (with respect
to either right or left uniform structure on G).Let be a compactly supported
aperiodic and centered probability measure on G.And let be its associated gaussian
distribution.Then we have
lim
n!+1
Z
f(g)d
n
(g)
Z
f(g)d
n
(g)
= 0
Proof.We may assume kfk
1
1.Fix"> 0 and let!> 0 be a modulus of
continuity for f relatively to",i.e.jf(ux) f(x)j "if kuk !and x 2 G,
where kgk = maxfjxj;jyj;jzjg for g = (x;y;z) 2 G.As follows from the central limit
theorem,we can nd a number C > 0 such that if we let A
n
= fg = (x;y;z) 2
G;jxj;jyj C
p
n and jzj Cng,we have for large n
n
(A
cn
) "
and
n
(A
cn
) "
We can then nd a cover B
n
of the cube A
n
by less than O(n
2
=!
4
) disjoint translates
R
!
h of a small cube of the form R
!
= d
!
(R) where h 2 d
!
(G(Z)),d
!
is the dilation
on G with coecient of contraction!,and R is a fundamental domain for the co
compact lattice G(Z) in G.Now we can write
Z
fd
n
Z
fd
n
Z
B
n
fd
n
Z
B
n
fd
n
+2"
X
i
f(h
i
) j
n
(Rh
i
)
n
(Rh
i
)j +4"
O(
n
2
!
4
) sup
h2G
j
n
(Rh)
n
(Rh)j +4"
Here we can apply the uniform local limit theorem (Theorem 1.2) and get
lim
n!1
n
2
sup
h2G
j
n
(Rh)
n
(Rh)j = 0
Thus,we obtain the desired result.
The proof of Corollary 1.4 now follows immediately from the combination of the
last two lemmas.
10.2.Unipotent randomwalks and equidistribution on homogeneous spaces.
Here we shall conclude this paper and give a proof of Theorem 1.5.
Let G be a connected real Lie group and a lattice in G,that is,a discrete
subgroup of G such that the homogeneous space G= bears a nite Borel measure
invariant by the left action of G.An element u 2 G is called Adunipotent,when
the automorphism Ad(u) 2 GL(g) of the Lie algebra g of G is unipotent,i.e.every
eigenvalue of Ad(u) equals 1.A subgroup U G is called Adunipotent or simply
44 E.BREUILLARD
unipotent is every element u 2 U is Adunipotent.The action of U on G= is called
a unipotent ow.
In the early nineties,in a series of papers (see [Rat13]),M.Ratner proved the
validity in full generality of the RaghunathanDani conjectures for the action of
connected Adunipotent subgroups on G=.These results have had a number of
far reaching applications (some of which were obtained earlier by proving special
cases of the conjecture,like in Margulis'proof of the Oppenheim conjecture (1986)),
especially to number theory and lattice points counting problems (see the recent
survey [Bab]).The results,can be summarized as follows.First,if U is a connected
Adunipotent subgroup of G,for every x 2 G=,the orbit Ux has a\nice algebraic"
closure,that is,there exists a closed subgroup H G such that
Ux = Hx is closed
and bears a unique Hinvariant probability measure m
x
.Secondly,every Uergodic
probability measure on G= is of the form m
x
for some x 2 G=.We refer the reader
to the surveys [Rat3] and [Sta] for a detailed exposition of these results and further
references (see also [MaT] for an alternative proof).
One of the main steps in the proof of the latter conjecture is the following equidis
tribution theorem for the action of oneparameter unipotent ows:
Theorem 10.3 (M.Ratner).Suppose G is a Lie group and a lattice in G.Let
U = fu(t);t 2 Rg be a oneparameter Adunipotent subgroup of G.Then for any
x 2 G=,there is a closed subgroup H of G,such that Hx is closed and bears an
Hinvariant probability m
x
,and the orbit Ux is equidistributed in Hx with respect to
m
x
.In other words,for all continuous and bounded functions f on G=,we have
lim
T!+1
1
T
Z
T
0
f(u(t)x)dt =
Z
Hx
fdm
x
Let us emphasize the fact that this equidistribution holds for every point x 2 G=
and not only almost everywhere with respect to some Uergodic measure.
10.2.1.A counterexample.In this paragraph,we illustrate the phenomenon de
scribed in the introduction of a noncentered random walk that may diverge in G=.
More precisely,we have:
Proposition 10.4.Let G = SL
2
(R), = SL
2
(Z) and U be a oneparameter unipo
tent subgroup of G.Let be a noncentered probability measure on U with standard
deviation < +1 and average d 6= 0.Then for any compact subset K in G= and
almost all points x 2 G= ( is a Ginvariant measure on G=),we have
(67) lim inf
n!+1
n
x
(K) = 0
Let S
n
= X
1
+:::+ X
n
be a sum of centered i.i.d.variables on the line with
standard deviation > 0:Then the random walk (S
n
+nd)
n
is noncentered with
drift d,and we can choose X
1
so that the probability law of S
n
+nd is precisely
n
.
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 45
We identify G= with the space of lattices in R
2
and we let kk be the standard
Euclidean norm on R
2
.Recall,that according to Mahler's criterion,a subset K
G= is relatively compact if and only if there exists > 0 such that kvk > for any
lattice x 2 K and any nonzero vector v 2 x.We set x
0
= Z
2
:
Let (u(t))
t
be a oneparameter unipotent subgroup in SL
2
(R).It stabilizes a line
D in R
2
and we let 2 R be the slope (that we assume nite) of this line.In the
canonical coordinates of R
2
;the action of u(t) reads:
u(t)
xy
=
x +t(y x)
y +t(y x)
for some 2 Rnf0g.
We say that a real is well approximable on both sides if for any"> 0 and
2 f1;1g one can nd integers x and y in Z
2
nf(0;0)g such that
jx(y x)j <"and x(y x) > 0
For the Lebesgue measure on R,almost all are well approximable on both sides.
Let us x an arbitrary compact subset K G= and let > 0 be small enough
so that kvk > for any lattice x 2 K and any nonzero vector v 2 x:Let
=
ft;u(t)x
0
=2 Kg.Then we claim:
Claim 10.5.If =2 Q is well approximable on both sides,then for any C > 0 and
n
0
> 0 there is an integer n > n
0
such that
S
n
+nd 2
whenever jS
n
j < C
p
n.
Proof of claim:For (x;y) 2 Z
2
nf(0;0)g we set A
x;y
= ft 2 R;ju(t)(x;y)j < g and
= j2j maxfjj;1g and t
x;y
=
x
(xy)
,I
x;y
= ft;jtj <
jxyj
g so that
[
x;y2Z
2
nf(0;0)g
A
x;y
If is small enough,since =2 Q is well approximable on both sides,one can nd
arbitrarily large integers x;y such that
(68) jx(y x)j <
4
and t
x;y
has the sign we want.Note that if is small enough,then t
x;y
+I
x;y
A
x;y
.
We can also assume jI
x;y
j > 2jdj.Hence,for some integer n,
(69) nd 2 t
x;y
+
1
2
I
x;y
Now if S
n
is not too large,i.e.if jS
n
j <
1
4
jI
x;y
j,then S
n
+nd 2 A
x;y
.On the
other hand,if n satises (69) then nd
2jxj
jjjxyj
:Thanks to (68),it follows that
jS
n
j <
1
4
jI
x;y
j whenever jS
n
j < C
p
n,with C
=
1
p
jdj=2
2
:Taking a smaller is
necessary,we obtain the desired conclusion.
46 E.BREUILLARD
Now the proof of the proposition follows easily from the central limit theorem:if
"> 0 one can nd C > 0 so that for all large enough n;P(jS
n
j < C
p
n) 1 ".
Hence;
(70) limsupP(u(S
n
+nd)x
0
=2 K) = 1
We thus have established (67) for x = x
0
and as soon as the slope of the line
D xed by U satises the diophantine condition dened above.If E is the set of
g 2 G such that (67) holds for x = g
1
x
0
;then E contains the set of g 2 G such that
the slope of gD is irrational and well approximable on both sides.The map from G
to R which sends g to the slope of gD has no critical points (it identies with left
translation on G=P).Hence E is a set of full Haar measure,and this ends the proof
of the proposition.
Remark 10.6.The same idea shows that a noncentered random walk can stay very
close to a closed orbit of U at arbitrary large time and with high probability.In
any case,it prevents the walk to equidistribute in the closure of the orbit.A similar
phenomenon arises as soon as the action of U on G= is not uniquely ergodic,i.e.
if U stabilizes a proper homogeneous subspace of G=.
10.2.2.Proof of Theorem 1.5.Making use of the equidistribution theorem above
(Theorem 10.3),N.Shah (cf.[Sha]) subsequently extended this result to the action
of an arbitrary simply connected Adunipotent subgroup U of G.Let us introduce
N.Shah's result.
Let U be any simply connected nilpotent Lie group and (v
1
;:::;v
k
) be a basis of
the Lie algebra u of U.This basis is called a triangular basis (or strong Malcev
basis) if the subspaces spanned by (v
i
;:::;v
k
) for any i are ideals of u,that is [v
i
;v
j
] 2
span(v
m
;:::;v
k
) where m = maxfi;jg + 1.Such a basis gives rise to polynomial
coordinates on U,i.e.the map
:R
k
!U
(t
1
;:::;t
k
) 7!exp(t
k
v
k
) ::: exp(t
1
v
1
)
is polynomial dieomorphism.It also sends the Lebesgue measure on R
k
to the Haar
measure on U.With this terminology,Shah proved (cf.[Sha] Cor.1.3.)
Theorem 10.7 (N.Shah).Suppose G is a Lie group and a lattice in G.Let U be
a simply connected Adunipotent subgroup of G.Let (v
1
;:::;v
k
) be a triangular basis
for U,and x 2 G=.Then for any continuous and bounded function f on G=,
lim
T
1
!1;:::;T
k
!1
1
T
1
:::T
k
Z
[0;T
1
]:::[0;T
k
]
f((t
1
;:::;t
k
)x)dt
1
:::dt
k
=
Z
Hx
fdm
x
where m
x
is the Hinvariant probability measure on
Ux = Hx.
This theorem is precisely what we need to apply Corollary 1.4 to the situation
of Theorem 1.5.Keeping the notations of the statement of Theorem 1.5,let f be
LOCAL LIMIT THEOREMS AND EQUIDISTRIBUTION ON THE HEISENBERG GROUP 47
a compactly supported function on G=,and suppose that U is isomorphic to the
Heisenberg group with triangular basis given by (3) in section 2.Then the function
F(u) = f(ux) = f((u
x
;u
y
;u
z
)x) is a bounded uniformly continuous function for
the left uniform structure on U satisfying the condition of Corollary 1.4 with limit
`=
R
Hx
fdm
x
.Therefore this is the end of the proof and of this paper.
Acknowledgments 10.8.This work is part of the author's Yale University Ph.D.
thesis.I sincerely thank my supervisor Gregory Margulis for suggesting this problem
and oering his invaluable help and guidance throughout the last few years.I am
also very grateful to Yves Guivarc'h for our long discussions and to Martine Babillot
whose inspiring enthusiasm I now miss dearly.
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Emmanuel Breuillard,IHES,Le BoisMarie,35 route de Chartres,F91440 Bures
surYvette,FRANCE
Email address:emmanuel.breuillard@ihes.fr
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