# Pairs of saddle connections of typical flat surfaces on fixed affine
orbifolds.

Bonnafoux  Centre de mathématiques Laurent-Schwartz, Ecole polytechnique,
91128 Palaiseau Cedex, France  Etienne
email<EMAIL_ADDRESS>Centre de mathématiques Laurent-
Schwartz, Ecole polytechnique, 91128 Palaiseau Cedex, France

###### Abstract

We prove that the asymptotic number of pairs of saddle connections with length
smaller than $L$ with bounded virtual area is quadratic (with power saving
error term) for almost every translation surface with respect to any ergodic
$SL(2,\mathbb{R})$-invariant measure. A key tool of the proof is that Siegel-
Veech transforms of bounded functions with compact supports are in
$L^{2+\kappa}$ for every $SL(2,\mathbb{R})$-invariant measure.

Key words: Saddle connection, Translation surface, Counting

2020 Mathematics Subject Classification: 32G15, 30F30 (Primary); 28C10
(Secondary)

##  1 Introduction

Given a translation surface $(X,\omega)$, that is a pair made of a Riemann
surface $X$ and a non-zero holomorphic one form $\omega$, a saddle connection
is defined as a flat geodesic between two zeros of $\omega$. All along this
article we might write $\omega$ for a translation surface for succinctness.

To each saddle connection $\gamma$ is associated its holonomy vector

$z_{\gamma}=\int_{\gamma}\omega\in\mathbb{C}.$

The length of $\gamma$ is then defined as $|z_{\gamma}|$.

We will count pairs of saddle connections with constraint length and _virtual
area_ which is defined by the equation:

$|z_{\beta}\wedge
z_{\gamma}|=|\operatorname{Re}(z_{\beta})\operatorname{Im}(z_{\gamma})-\operatorname{Im}(z_{\beta})\operatorname{Re}(z_{\gamma})|$

for two saddle connections $\gamma$ and $\beta$.

For $\omega$ a translation surface, we call $\Lambda_{\omega}$ the set in
$\mathbb{C}$ of all holonomy vectors of the saddle connections in $\omega$.
This is a discrete locally finite set.

The moduli space $\Omega_{g}$ of compact genus $g$ area $1$ translation
surfaces, is stratified by integer partitions of $2g-2$, and each stratum is
divided in different connected components. For the rest of the paper we fix a
genus $g$, an integer partition and one of its connected component
$\mathcal{H}$[9].

On $\Omega_{g}^{\prime}$ the moduli space of compact genus $g$ without area
restriction, there is a natural measure $\mu_{MV}^{\prime}$ called Masur-Veech
measure which is locally the Lebesgue measure with respect to the period
coordinate [10, 15]. Then by coning, a measure $\mu_{MV}$, also named Masur-
Veech measure, is defined on $\Omega_{g}$, that is for every $A$ measurable:

$\mu_{MV}(A)=\mu_{MV}^{\prime}(\\{s\omega,0\leq s\leq 1,\omega\in A\\}).$

###  1.1 History and prior results

The saddle connections are keystones for studying the geometry of flat
surfaces. In fact a result of Masur and Smillie [12] says that the compact
part of $\Omega_{g}$ is made of surfaces with no saddle connection of length
less than a given $\epsilon>0$.

Concerning the counting of these objects, Masur [11] managed to bound the
function $N(\omega,R)=\Lambda_{\omega}\cap B(0,R)$. More precisely, he showed
that for every $\omega\in\mathcal{H}$ , there are $c_{1}(\omega)$ and
$c_{2}(\omega)$ such that

$c_{1}(\omega)R^{2}\leq\\#N(\omega,R)\leq c_{2}(\omega)R^{2}.$

Later on, Veech [16] found an $L^{1}$-quadratic asymptotic formula: namely,
there is a $c>0$ such that for $\mu_{MV}$-almost every $\omega$,

$\underset{R\to\infty}{\lim}\int_{\mathcal{H}}\left|\frac{N(\omega,R)}{R^{2}}-c\right|d\mu(\omega)=0$

This result was subsequently improved by Eskin and Masur [6] who showed that,
for every $\mu$ an ergodic $SL(2,\mathbb{R})$-invariant measure, there is a
$c>0$ such that for $\mu$-almost every $\omega\in\mathcal{H}$

$\underset{R\to\infty}{\lim}\frac{N(\omega,R)}{R^{2}}=c.$

More recently Athreya, Fairchild and Masur [2] extended this result to a
counting of pairs of saddle connections with a constraint on the virtual area.
More concretely, consider the following counting function

$N_{A}(\omega,R)=\\#\\{(z,w)\in\left(\Lambda_{\omega}\cap
B(0,R)\right)^{2}:|z\wedge w|\leq A,|w|\leq|z|\\}.$

######  Theorem 1.1.

There is a constant $c(A)$ such that for $\mu_{MV}$-almost every
$\omega\in\mathcal{H}$

$\underset{R\to\infty}{\lim}\frac{N_{A}(\omega,R)}{R^{2}}=c(A).$

The main goal of this paper is to generalize this result to other
$SL(2,\mathbb{R})$-invariant measures.

###  1.2 General $SL(2,\mathbb{R})$-invariant measures

The group $SL(2,\mathbb{R})$ acts on $\Omega_{g}$ and preserve $\mu_{MV}$
(which is ergodic with respect to its action). We will write $g_{t}$ for its
diagonal action and $r_{\theta}$ for the rotation.

In general, a connected component of a stratum $\mathcal{H}$ carries many
$SL(2,\mathbb{R})$-invariant measures. As it was famously shown by Eskin and
Mirzakhani [7], such measures are supported on affine orbifolds, here denoted
by $\mathcal{M}$. Roughly speaking, this mean that in any point
$p\in\mathcal{M}$, there is a neighborhood $U$ such that $\mathcal{M}\cap U$
is map by local period coordinates on a subspace defined by real linear
equations.

Our result is that theorem 1.1 also holds for this family of measures:

######  Theorem 1.2.

Given $\mu$ an ergodic $SL(2,\mathbb{R})$-invariant measure, there is a
$c_{\mu}(A)$ and a $\kappa>0$ such that for $\mu$-almost every
$\omega\in\mathcal{H}$

$\frac{N_{A}(\omega,R)}{R^{2}}=c_{\mu}(A)+O_{\omega}(R^{-\kappa}).$

Furthermore if a sequence of probability $SL(2,\mathbb{R})$-invariant ergodic
measure $\mu_{n}$ weakly converge to a probability
$SL(2,\mathbb{R})$-invariant measure ergodic $\mu$, then for any $A>0$

$c_{\mu_{n}}(A)\to c_{\mu}(A)$

The proof of this result occupies the rest of this paper. More precisely we
begin by showing in Section 2 and 3 that

$\underset{R\to\infty}{\lim}\frac{N_{A}(\omega,R)}{R^{2}}=c_{\mu}(A).$

Our argument will follow closely the proof of Athreya, Fairchild and Masur,
with three significant changes, as indicated in Subsection 1.4 below. Then in
Section 4 we will derive the power saving error term
$O_{\omega}(R^{-\kappa})$, based on tools developed by Nevo, Rühr and Weiss
[14]. Finally, continuity properties of the constants $c_{\mu}(A)$ are
discussed in Section 5.

For our purpose, it is important to recall that there is a notion of
independence for the saddle connection attached to each affine orbifold
$\mathcal{M}$.

######  Definition 1.1.

Saddle connections $s_{1},\cdots,s_{k}$ on a surface $\omega\in\mathcal{M}$
are said to be $\mathcal{M}$ _-independent_ if their relative homology classes
define linearly independent functionals (over $\mathbb{C}$) on the linear
subspace $T_{\omega}\mathcal{M}\subset T_{\omega}\mathcal{H}\cong
H^{1}(\omega,\Sigma;\mathbb{C})$ where $\Sigma$ stand for the set of $0$ of
$\omega$.

###  1.3 Siegel-Veech transform

As remarked by Athreya, Fairchild and Masur in Section 1.4.3 of [2] one key
point to show theorem 1.2 is to extend a result on the integrability of the
Siegel-Veech transform of bounded compact functions which we recall now.

Let $B_{c}(X)$ be the space of bounded measurable functions with compact
support on a space $X$. The Siegel-Veech transform for a function $f\in
B_{c}(\mathbb{R})$ is defined as

$\hat{f}(\omega)=\sum_{z\in\Lambda_{\omega}}f(z).$

Sometimes for readability, we will also use the notation $f^{SV}$.

In this setting Athreya, Cheung and Masur [1] proved that:

######  Theorem 1.3.

There is a $\kappa>0$ such that for every $f\in B_{c}(\mathbb{C})$,
$\hat{f}\in L^{2+\kappa}(\mathcal{H},\mu_{MV})$.

As it turns out, our goal in the next section, will be to show the following
extension of Theorem 1.3.

######  Theorem 1.4.

For every $SL(2,\mathbb{R})$-invariant measure $\mu$, there is a $\kappa>0$
such that for every $f\in B_{c}(\mathbb{C})$, $\hat{f}\in
L^{2+\kappa}(\mathcal{H},\mu)$.

As it is discussed in the beginning of Section 2.2 of [2], this theorem
extends to function of two variables. For a function $g\in
B_{c}(\mathbb{R}^{2})$, the Siegel-Veech transform is defined as

$\hat{g}(\omega)=\sum_{z_{1},z_{2}\in\Lambda_{\omega}^{2}}g(z_{1},z_{2}).$

An easy corollary of the previous theorem is

######  Corollary.

For every $\mu$ $SL(2,\mathbb{R})$-invariant measure, there is a
$\kappa^{\prime}>0$ such that for every $h\in B_{c}(\mathbb{R}^{2})$,
$\hat{h}\in L^{1+\kappa^{\prime}}(\mathcal{H},\mu)$.

###  1.4 Review of the proof of Theorem 1.1

As our argument towards Theorem 1.2 follow closely the proof of Theorem 1.1,
we will give a quick landscape on it and comment on the points which need to
be adapted.

To begin, Athreya, Fairchild and Masur focus on partial counting functions

$N^{*}_{A}(\omega,R)=N_{A}(\omega,R)-N_{A}(\omega,R/2).$

These functions count the number of pairs of saddle connections such that
their period coordinates are in

$D_{A}(R,R/2):=\left\\{(z,w)\in\mathbb{C}^{2}:R/2<|z|<R,|w|<|z|,|w\wedge
z|\leq A\right\\}$

Estimates on these function can be extended to $N_{A}(\omega,e^{t})$ using
geometric series argument with a suitable control on a upper bound on
$N^{*}_{A}(\omega,R)$ given by Proposition 3.2 in their article.

Then they describe a set

$R_{A}(\mathcal{T}):=\left\\{(z,w)\in\mathbb{C}^{2},1/2\leq\operatorname{Im}(z)\leq
1,|\operatorname{Re}(z)|\leq\operatorname{Im}(z),|\operatorname{Im}(w)|\leq\operatorname{Im}(z),|w\wedge
z|\leq A\right\\}$

called the fibered trapezoid, which have the property that its characteristic
function $h_{A}$ satisfies

$|N^{*}_{A}(\omega,R)-\pi
e^{2t}(A_{t}\hat{h}_{A})(\omega)|=|m_{t}(\omega)+\sum_{i=1}^{4}e^{i}_{t}(\omega)|$
(1)

where $A_{t}$ is the following averaging operator

$A_{t}(h)(p)=\frac{1}{2\pi}\int_{0}^{2\pi}h(g_{t}r_{\theta}p)d\theta$

and $m_{t}$ is called _main term_ and the $e^{i}_{t}$ _error terms_.

These functions are defined as the difference of the two terms in the left
hand side of equation (1) estimated on various loci. The definitions of these
loci are the following

  * •

_Main part_

$M_{t}=\left\\{(z,w)\in
D_{A}\left(\sqrt{\frac{cosh(2t)}{2}},e^{t}\right):|w|<|z|(1+e^{-4t})^{-1/2}\right\\},$

  * •

_Bottom of the trapezoid_

$E^{1}_{t}=D_{A}\left(e^{t}/2,\sqrt{\frac{cosh(2t)}{2}}\right),$

  * •

_The norm of $w$ is greater than $|z|(1+e^{-4t})^{-1/2}$_

$E^{2}_{t}=\left\\{(z,w)\in
D_{A}\left(\sqrt{\frac{cosh(2t)}{2}},e^{t}\right):|w|>|z|(1+e^{-4t})^{-1/2}\right\\},$

  * •

_Top of the trapezoid_

$E^{3}_{t}=\left\\{(z,w)\in\mathbb{C}^{2}:(A_{t}h_{A})(z,w)>0,|z|>e^{t}\right\\},$

  * •

_The averaging operator is positive but_ $(z,w)\notin D_{A}(e^{t}/2,e^{t})$

$E^{4}_{t}=\left\\{(z,w)\in\mathbb{C}^{2}:(A_{t}h_{A})(z,w)>0,e^{t}/2<|z|<e^{t},|w|>|z|\right\\}.$

After that the error functions are defined as

$e^{i}_{t}(\omega)=(\chi_{E^{i}_{t}}\cdot(\chi_{D_{A}(e^{t}/2,e^{t})}-\pi
e^{2t}(A_{t}h_{A})))^{SV}(\omega)$

and

$m_{t}(\omega)=(\chi_{M_{t}}\cdot(\chi_{D_{A}(e^{t}/2,e^{t})}-\pi
e^{2t}(A_{t}h_{A})))^{SV}(\omega).$

where $\chi_{\cdot}$ are characteristic functions (this notation will be used
all along this article).

To control the limit of $(A_{t}\hat{h}_{A})$ as $t\to\infty$, they use Nevo
ergodic theorem [13] stating that

######  Theorem 1.5.

Suppose $\mu$ is an ergodic $SL(2,\mathbb{R})$-invariant probability measure
on $\mathcal{H}$. Assume $f\in L^{1+\kappa}(\mathcal{H},\mu)$ for some
$\kappa>0$, and that $f$ is $K$-finite, that is,
$f_{\theta}(\omega):=f(r_{\theta}\omega)$ the span of the functions
$\\{f_{\theta}:\theta\in[0,2\pi[\\}$ is finite-dimensional. Let $\eta\in
C_{c}(\mathbb{R})$ be a continuous non negative bump function with compact
support and of unit integral. Then for $\mu$-almost every
$\omega\in\mathcal{H}$,

$\underset{t\to\infty}{\lim}\int_{-\infty}^{\infty}\eta(t-s)(A_{s}f)(\omega)ds=\int_{\mathcal{H}}fd\mu$

Using this theorem they get a result on the limit of the averaging operator.

######  Theorem 1.6 (Theorem 2.2 of [2]).

For $\phi\in C_{c}(\mathbb{C}^{2})$, for $\mu_{MV}$-almost every
$\omega\in\mathcal{H}$, the circle average of $\hat{\phi}$ converge

$\underset{t\to\infty}{\lim}A_{t}\hat{\phi}(\omega)=\int_{\mathcal{H}}\hat{\phi}d\mu_{MV}$

To adapt this theorem to our case of an arbitrary $SL(2,\mathbb{R})$-invariant
ergodic measure $\mu$, we need to show that there is a $\kappa>0$ such that
for any $f\in B_{c}(X)$, $f\in L^{2+2\kappa}(\mathcal{H},\mu)$ which will
imply that for any $h\in B_{c}(\mathbb{C}^{2})$, $\hat{h}\in
L^{1+\kappa}(\mathcal{H},\mu)$. For the Masur-Veech measure, this statement is
the main result of [1]. In our setting, this is the main content of Section 2.

Once this is done, taking a bump function $g_{\epsilon}$ (whose existence is
established in Lemma 3.4 of [2]) such that

$|A_{t}\hat{h}_{A}(\omega)-A_{t}\hat{g}_{\epsilon}(\omega)|\leq\epsilon$

and

$|\int_{\mathcal{H}}\hat{g}_{\epsilon}-\hat{h}_{A}d\mu|\leq\epsilon$

we get that

$|A_{t}\hat{h}_{A}(\omega)-\int_{\mathcal{H}}\hat{h}_{A}d\mu|\leq|A_{t}\hat{h}_{A}(\omega)-A_{t}\hat{g}_{\epsilon}(\omega)|+|A_{t}\hat{g}_{\epsilon}(\omega)-\int_{\mathcal{H}}\hat{g}_{\epsilon}d\mu|+|\int_{\mathcal{H}}\hat{g}_{\epsilon}-\hat{h}_{A}d\mu|\leq
3\epsilon$

for $t$ big enough, using the extension for $\mu$ of Theorem 1.6 on
$A_{t}\hat{g}_{\epsilon}$ for the second term.

It remains to prove that for almost every $\omega$,

$|m_{t}(\omega)|=o(e^{2t})$ (2)

and for $i=1,2,3,4$

$|e^{i}_{t}(\omega)|=o(e^{2t}).$ (3)

For this sake, in the case of the Masur-Veech measure, Athreya, Fairchild and
Masur started by estimating the volume of the set with two non-homologous
small saddle connections (cf. Lemma 5.2 of [2]). In our current setting this
lemma has an equivalent form thanks to Dozier [5].

Next, Athreya, Fairchild and Masur have two lemmas (cf. Lemmas 5.3 and 5.4 of
[2]) bounding the function $N(\omega,L)$. As they show in their paper, these
lemmas essentially follow paragraph 3.6.2 and 3.6.3 of [1] and [6]. Since they
are stated for every $\omega\in\mathcal{H}$, we don’t need to modify them
here.

Afterwards, Athreya, Fairchild and Masur bound (cf. Lemma 5.5 of [2]) the
Masur-Veech integral of a counting function in the thin part of the moduli
space (where some saddle connections are small). This mostly uses the estimate
of Lemma 5.2 (which as it was previously said is extended by Dozier’s result
[5] and can be used to have similar result of Lemma 5.2) except a couple of
technical terms should be adapted to be extended to any
$SL(2,\mathbb{R})$-invariant measure.

Furthermore, Lemmas 5.6 and 5.7 of [2] are technical lemmas of a geometric
nature, controlling the orbit of a point in $\mathbb{C}$ under geodesic flow
and rotation: this means that we can still use it.

Moreover, they need to control the Masur-Veech volume of different loci on the
thick part of the stratum related to the error terms (cf. Lemma 5.9 of [2]).
In the case of an arbitrary $SL(2,\mathbb{R})$-invariant ergodic measure, we
shall need to add extra arguments. This will be done in Section 3.

In the end, using all these facts, we can follow in a straightforward way the
same computation of Athreya, Fairchild and Masur in Section 5.9 of [2] to get
the bounds on the main term (2) and the error terms (3) and _a fortiori_
combining the different limits the desired result.

In summary, our task for proving the quadratic asymptotics in Theorem 1.2 is
reduced to show the integrability of the Siegel-Veech transform with respect
to any $SL(2,\mathbb{R})$-invariant ergodic measure (cf. Theorem 1.4) which
will be done in Section 2 and two adapt three lemmas (Lemma 5.2, 5.5 and 5.9
of [2]) which will be the content of Section 3.

###  1.5 Compactification of the moduli space

A crucial tool to show Theorem 1.4 is the multi-scale compactification of
strata introduced by Bainbridge, Chen, Grushevsky and Möller [3].

Indeed, Dozier uses it to bound the volume of a set with a family of
$\mathcal{M}$-independent small saddle connections. Similarly, we want an
estimate on the volume of the set where two saddle connections are small and
in some precise position. Namely, one is a circumference curve of a cylinder
and the other one cross the same cylinder. This set appears in the proof of
the integrability of the Siegel-Veech transform, and in some sens, the control
of his measure is one of the novelty of this paper.

Let us recall quickly the compactification process. A multi-scale differential
is the data of

  * •

A nodal Riemann surface $M$,

  * •

A graph, where each vertex correspond to a component of the surface $M$, and
the edge to a node between the surfaces,

  * •

Half-edge recording in which component the $0$ of the differential are,

  * •

A level function $l$ assigning a non positive integer to each vertex. We take
it to be surjective in $[-N,0]\cap\mathbb{Z}$ for some $N$. The union of
component with the same level, that is $l^{-1}(i)$ for a given $i$ is called a
_level subsurface_ and denote by $X^{(i)}$,

  * •

A positive integer $b_{e}$ to each vertical edge which record the cone angle,

  * •

A collection of meromorphic differentials on each component of the surface
which are consistent with the previous data (we won’t give here the meaning of
this term, but we refer the reader to [3] for full definition),

  * •

And a prong-matching record, giving the identification of the horizontal
direction between two sides of a node.

The compactified stratum is denoted $\bar{\mathcal{H}}$. On it, special
neighborhoods are handy for computation. They are described by the following
definition and theorem.

######  Definition 1.2.

A connected, open subset $Q\subset\bar{\mathcal{H}}$ is said to be a _period
coordinate chart_ if it admits an injective map to $\mathbb{C}^{n}$ that is
locally linear with respect to the period coordinates.

Fix $\bar{X}$ in the boundary, there is a complex-analytic coordinate in which
we can define a good neighborhood.

  * •

In each component, there are _moduli parameters_ $s_{i}\in\mathbb{C}$ that we
take small, that is $0<|s_{i}|<\epsilon$,

  * •

For each node, there is a _smoothing parameter_ $t\in\mathbb{C}$. We also take
them with a restriction on the norm $0<|t|<\epsilon$,

  * •

If $t$ is a horizontal node parameter, then we consider a restriction of the
smoothing parameter

$arg(t)\in(\alpha,\alpha+\pi/4)$

where we choose finitely many $\alpha$ such that the family of such intervals
cover all the circle,

  * •

If $t_{i}$ is a scaling parameter for level $i$, with associate integer
$a_{i}$, we find connected interval condition on $arg(t_{i})$ such that
$arg(t_{i}^{a_{i}})$ satisfies the same condition as in the previous point.

Dozier show in Lemma 3.6 of [5] that:

######  Theorem 1.7.

The previously defined neighborhoods are period coordinate charts, for
$\epsilon$ small enough.

###  1.6 Ordering of level subsurfaces

The ordering given by the level function might not be a total order. To
structure the surface, we should make other choices.

Let $\mathcal{S}$ be the set made of level subsurfaces and degenerating
cylinders. Dozier defined a function
$size_{\omega}(\cdot):\mathcal{S}\to\mathbb{R}$ for every $\omega\in V$ with
$V$ a neighborhood as in theorem 1.7.

Then, if $\mathcal{O}$ is the set of ordering on $\mathcal{S}$ that restricts
to the ordering given by the level function and have the property that if $C$
is a degenerating cylinder and $X^{i}$ is the level subsurface at which the
circumference curve of $C$ lies, then $C\succ X^{i}$.

We say that $\succ\in\mathcal{O}$ is _consistent_ with $X\in V$ if

$size_{\omega}(Y_{1})\geq size_{\omega}(Y_{2})\implies Y_{1}\succ Y_{2}.$

Finally, we call a degenerating cylinder $\succ$_-wide_ if it is
$\succ$-greater than any level subsurface.

###  1.7 Delaunay triangulation

One last tool that we will use in Section 3 is Delaunay triangulation. Let’s
consider the _Voronoi decomposition_ of a translation surface with each cell
centered around singular point. Then the dual of this decomposition is the
_Delaunay decomposition_. Further decompositions are called _Delaunay
triangulation_. Refer to [12] for complete definition.

One key point of these triangulations is the so-called efficiency property

######  Theorem 1.8.

Let $S$ be a translation surface and $T$ a Delaunay triangulation. Then every
saddle connection $\beta$ is homotopically equivalent to a path $P(\beta)$ in
the Delaunay triangulation whose length satisfies

$|P(\beta)|\leq\sqrt{10}|\beta|.$

###### Proof.

See Lemma 2.3 of [1]. ∎

###  1.8 Acknowledgements

I would like to thank Benjamin Dozier for his crucial remarks which greatly
improve this paper. I am thankful to Jon Chaika for pointing me out the result
of Dozier on convergence of Siegel-Veech constants. I am also grateful to
Jayadev Athreya, Samantha Fairchild and Howard Masur for inspiring discussion
at CIRM during the Combinatorics, Dynamics and Geometry on Moduli Spaces
conference. Finally, I would like to thank Carlos Matheus for introducing me
to the subject and advising me all along.

##  2 Siegel-Veech transforms are in $L^{2+2\kappa}(\mathcal{H},\mu)$

In this section, we now prove Theorem 1.4.

We use the same strategy as in [1] by starting with the characteristic
function of the disc of radius a small $\epsilon_{0}$ centered at $0$.

######  Theorem 2.1.

Let $\epsilon_{0}>0$ and let $f$ be the characteristic function of the disc of
radius $\epsilon_{0}$ centered at $0$. Then there is a $\kappa>0$ such that
$\hat{f}$ is in $L^{2+2\kappa}(\mathcal{H},\mu)$.

######  Remark.

$\epsilon_{0}$ should be taken small enough later, in the last paragraph.

We discuss the integrability of this function on several loci of
$\mathcal{H}$. For all of them, but the last one, the arguments are the same
as in [1]. We will briefly remind them to concentrate on the last one.

##### Thick part

On the locus with no saddle connection with length smaller than
$\epsilon_{0}$, we have $\hat{f}=0$.

##### No short loops

On the set where there is a saddle connection of length less than
$\epsilon_{0}$, but no homotopic nontrivial closed curves of length less than
$\epsilon_{0}$, following paragraph 3.4 of [1], $\hat{f}$ is bounded on this
set. As the measure $\mu$ is of finite volume, this concludes this case.

##### Short loops

On the case where there are short loops of length smaller than $\epsilon_{0}$,
we will subdivide into four cases
$\Omega_{0},\Omega_{1},\Omega_{2},\Omega_{3}$.

On each, we want to show that

$\sum_{k\geq 1}\mu(\Omega_{i}(k))<+\infty$

where $\Omega_{i}(k):=\\{\hat{f}\geq k^{1/q}\\}\cap\Omega_{i}$, for a $q>2$
that we will fix latter.

For the rest of this section we will call $\gamma$ the smallest saddle
connection and $|\epsilon|$ the length of the smallest saddle connection which
is non parallel with $\gamma$. Note that in the case of two saddle connections
non-parallelism implies $\mathcal{M}$-independence as explain in $\S$ 1.4 of
the article of Dozier [5].

Theorem 5.1 of [6] ensures that if $N$ is the dimension of relative homology,
for every $\delta<\frac{1}{N}$ there is a $C$ such that

$\hat{f}(\omega)\leq\frac{C}{|\gamma|^{1+\delta}}.$

We can deduce that if $\hat{f}\geq k^{1/q}$ there is a $c$ such that

$|\gamma|\leq ck^{-\frac{1}{q(1+\delta)}}.$

Choose for the rest of the section, any $\delta$ and $p$ such that

$0<\delta<p<\frac{1}{N}.$

We will fix them later.

##### The shortest non-parallel saddle connection is shorter that a power of
the shortest

Let

$\Omega_{0}(k)=\\{(X,\omega)\in\mathcal{H},\hat{f}(\omega)\geq
k^{1/q},|\epsilon|\leq|\gamma|^{p}\\}.$

The regularity property of the measure $\mu$ proved by Dozier [5], knowing
non-parallelism and hence the $\mathcal{M}$-independence of
$(\gamma,\epsilon)$, ensures that

$\mu(\Omega_{0}(k))=O(|\gamma|^{2+2p})=O(k^{-\frac{2(1+p)}{q(1+\delta)}}),$

which is summable if $q<2\frac{1+p}{1+\delta}$

##### The shortest non-parallel saddle connection is longer that a power of
the shortest and the shortest is not a cylinder curve

Let

$\Omega_{1}(k)=\\{(X,\omega)\in\mathcal{H},\hat{f}(\omega)\geq
k^{1/q},|\epsilon|\geq|\gamma|^{p},\gamma\textrm{ is not on the boundary of a
cylinder}\\}$

Let’s recall a lemma from Athreya, Cheung and Masur [1].

######  Lemma 2.2 (Lemma 3.2 of [1]).

Suppose $\gamma$ is the shortest saddle connection on $(X,\omega)$. Let
$\beta$ be a saddle connection (with an orientation) such that the path
$P(\beta)$ in the Delaunay triangulation given by Theorem 1.8 follows edges
parallel to $\gamma$ more than $2M+1$ times, where $M$ is the total number of
triangles in the Delaunay triangulation. Then there is a cylinder $C$ with
$\gamma$ on its boundary and $\beta$ crosses $C$.

As in this case $\gamma$ is not on the boundary of a cylinder, every saddle
connection $\beta$ such that the path $P(\beta)$ follow $2M$ times a edge
parallel with $\gamma$ must follow an edge non parallel with $\gamma$ after
that, and hence of length at least $|\epsilon|$. Thus any saddle connection
with length less that $\epsilon_{0}$ can be decomposed as $O(|\epsilon|^{-1})$
edges of the triangulation and hence decomposed in a basis of
$H_{1}(X,\omega,\Sigma)$ with coefficient that are $O(|\epsilon|^{-1})$. The
number of saddle connection smaller than $\epsilon_{0}$ is then
$O(|\epsilon|^{-N})$ where $N$ is the dimension of $H_{1}(X,\omega,\Sigma)$.

To summary

$\hat{f}(X,\omega)=O(|\epsilon|^{-N})=O(|\gamma|^{-Np}).$

The result recalled at the beginning of subparagraph "short loops" gives that,
if $\hat{f}$ has to be bigger than $k^{1/q}$, it is needed that

$|\gamma|=O(k^{\frac{-1}{qNp}}).$

Thus, using the regularity of the measure $\mu$,

$\mu(\Omega_{1}(k))=O(|\gamma|^{2})=O(k^{\frac{-2}{qNp}}),$

which is summable if $q<\frac{2}{Np}$.

##### The shortest non-parallel saddle connection is longer that a power of
the shortest and the shortest is a cylinder curve and the height of the
cylinder is at least $\epsilon_{0}$

Let

$\Omega_{2}(k)=\\{(X,\omega)\in\mathcal{H},\hat{f}(\omega)\geq
k^{1/q},|\epsilon|\geq|\gamma|^{p},\gamma\textrm{ is on the boundary of a
cylinder of height }\geq\epsilon_{0}\\}$

Here the same lemma as in the previous paragraph is applied, and we get that a
curve that follow $2M$ times edges parallel to $\gamma$ should after cross the
cylinder whose height is $\epsilon_{0}$. As the number of parallel edges is
bounded, so is $\hat{f}$. Then $\Omega_{2}(k)$ is empty for $k$ big enough.

##### The shortest non-parallel saddle connection is longer that a power of
the shortest and the shortest is a cylinder curve and the height of the
cylinder is at most $\epsilon_{0}$

Let

$\Omega_{3}(k)=\\{(X,\omega)\in\mathcal{H},\hat{f}(\omega)\geq
k^{1/q},|\epsilon|\geq|\gamma|^{p},\gamma\textrm{ is on the boundary of a
cylinder of height }\leq\epsilon_{0}\\}.$

For every point $X\in\bar{\mathcal{H}}$ choose a neighborhood which is a
period coordinate chart as in theorem 1.7. As the space is compact, we can
extract a finite covering. This family separates into two kinds, the first one
which are contained in the open locus of the compactification $\mathcal{H}$,
and the others which have a point on the boundaries
$\bar{\mathcal{H}}-\mathcal{H}$.

We fix $\epsilon_{0}$ small enough, such that $\Omega_{3}(k)$ is included in a
finite union of neighborhoods of the second type for every $k$.

Let $V$ be one of these neighborhoods. In this neighborhood we pick an
ordering $\succ$ of the subsurface. There is only a finite number of choice of
neighborhood and of ordering.

We are interested in the intersection of $V$ with $\Omega_{3}(k)$ and the set
of surfaces consistent with $\succ$. Let’s call this set
$B^{\succ}_{\Omega_{3}(k),V}$.

Let $\eta$ be a saddle connection included in the cylinder $C$, joining two
zeros on the boundaries of $C$ and crossing $\gamma$ only once. $\eta$ and
$\gamma$ are $\mathcal{M}$-independent since they are not parallel.

We add $\alpha_{1},\cdots,\alpha_{k^{\prime}}$, other saddle connections with
representatives leaving at level at most $X^{(0)}$ such that
$\gamma,\eta,\alpha_{1},\cdots,\alpha_{k^{\prime}}$ generate $H_{X^{0}\succ}$
the subspace of elements of $H_{1}(X,\Sigma;\mathbb{C})$ that lie at level
$X^{0}\succ$. They do not cross any $\succ$-wide cylinder.

Then for each $\succ$-wide cylinder, one by one, in the order given by
$\succ$, if the circumference curve is independent of the previous basis we
add a curve joining two zeros and crossing no other $\succ$-wide cylinder
(this is always doable). These saddle connections are called
$\alpha_{k^{\prime}+1},\cdots,\alpha_{m^{\prime}}$.

Then we can extract an $\mathcal{M}$-independent basis by removing, in the
order of apparition in
$\gamma,\eta,\alpha_{1},\cdots,\alpha_{k^{\prime}},\alpha_{k^{\prime}+1},\cdots,\alpha_{m^{\prime}}$,
the $\mathcal{M}$-dependent saddle connections. We get a new basis
$\gamma,\eta,\beta_{1},\cdots,\beta_{k},\beta_{k+1},\cdots,\beta_{m}.$

Next we bound the period coordinates of this basis. To do this we recall two
lemmas from Dozier [5].

######  Lemma 2.3.

For $\alpha$ a relative homology class defined on the surfaces in $V$, there
exists a constant $C>0$ with the following property. Fix $X\in V$ of area $1$
such that $\succ$ is consistent with $X$. Let $Y$ be the level of $\alpha$
with respect to $\succ$. Then

$|\alpha(X)|\leq Csize_{X}(Y).$

######  Lemma 2.4.

Let $\beta$ be a relative homology class defined on the surface in $V$ with
the following properties:

  * •

$\beta$ has a representative that crosses exactly one closed curve $\alpha$
that is the core curve of a degenerating cylinder. We let $Y$ be the level
subsurface containing $\alpha$.

  * •

$\beta$ has a representative such that the level subsurfaces which the
representative intersects all lie at or below the level of $Y$.

Then there exists a family of rectangle $\mathcal{R}(z)\subset\mathbb{C}$, of
area bounded above by some $R$ (depending on $V$ and $\beta$) with the
following property. For any $X\in V$ of area $1$, we have,

$\beta(X)\in\mathcal{R}(\alpha(X)).$

Furthermore, the rectangles have the property that
$s\mathcal{R}(z)\subset\mathcal{R}(sz)$ for any $z\in\mathbb{C}$ and $0<s\leq
1$.

Using this two lemmas and prior information on $\gamma$ and $\eta$, we can
state that:

  * •

The period coordinates of $\gamma$ live in a disk of radius
$ck^{-\frac{1}{q(1+\delta)}}$,

  * •

The period coordinates of $\eta$ live in a rectangle centered at the origin
with one side of length $|\gamma|$ and one side $\epsilon_{0}$, we call
$R(z_{\gamma},\epsilon_{0})$ this rectangle,

  * •

All $\beta_{i}$ with $1\leq i\leq k$, with lemma 2.3 are bound by

$|\beta_{i}(X^{\prime})|\leq Csize_{X^{0}}(X^{\prime})\leq K$

because the size of the top subsurface is bounded in $V$,

  * •

All $\beta_{i}$ with $k+1\leq i\leq m$ are in
$\mathcal{R}(\gamma_{i}(X^{\prime}))$ with $\gamma_{i}$ the circumference
curves of the $\succ$-wide cylinders they cross, and $\mathcal{R}(\cdot)$ the
rectangles given by lemma 2.4. There are linear functions $f_{i}$ such that
$\gamma_{i}=f_{i}(\gamma,\eta,\beta_{1},\cdots,\beta_{k})$ for all $k<i\leq
m$. Indeed $\gamma_{i}$ belong to a sub-surface $X^{i}$ below the level
$X^{0}$ and $H_{X^{0}\succ}$ is generated by
$\gamma,\eta,\beta_{1},\cdots,\beta_{k}$. We call $\mathcal{R}_{i}$ the
rectangle $\mathcal{R}(f_{i}(\gamma,\eta,\beta_{1},\cdots,\beta_{k}))$ .

We can now integrate using Fubini. To shorten the notation we call the period
coordinate of $\beta_{i}$ $z_{i}:=z_{\beta_{i}}$.

$\displaystyle\mu(\\{B^{\succ}_{\Omega_{3}(k),V}\\})$
$\displaystyle=\mu^{\prime}(\\{sX:0\leq s\leq 1,X\in
B^{\succ}_{\Omega_{3}(k),V}\\})$
$\displaystyle=Leb(\\{s(z_{\gamma},z_{\eta},z_{1},\cdots,z_{k})\in\mathbb{C}^{k+2},0\leq
s\leq 1,z_{\gamma}\in B(0,ck^{-\frac{1}{q(1+\delta)}}),$ $\displaystyle
z_{\eta}\in R(z_{\gamma},\epsilon_{0}),z_{i}\in B(0,K),1\leq i\leq
m,z_{j}\in\mathcal{R}_{j},k+1\leq j\leq m\\})$ $\displaystyle\leq
Leb(\\{s(z_{\gamma},z_{\eta},z_{1},\cdots,z_{k})\in\mathbb{C}^{k+2},0\leq
s\leq 1,sz_{\gamma}\in B(0,sck^{-\frac{1}{q(1+\delta)}}),$ $\displaystyle
sz_{\eta}\in R(sz_{\gamma},s\epsilon_{0}),sz_{i}\in B(0,K),1\leq i\leq
m,sz_{j}\in s\mathcal{R}_{j},k+1\leq j\leq m\\})$ $\displaystyle\leq
Leb(\\{(z_{\gamma},z_{\eta},z_{1},\cdots
z_{k})\in\mathbb{C}^{k+2},z_{\gamma}\in B(0,ck^{-\frac{1}{q(1+\delta)}}),$
$\displaystyle sz_{\eta}\in R(z_{\gamma},\epsilon_{0}),z_{i}\in B(0,K),1\leq
i\leq m,z_{j}\in\mathcal{R}_{j},k+1\leq j\leq m\\})$
$\displaystyle=\underset{B(0,ck^{-\frac{1}{q(1+\delta)}})}{\int}\underset{R(z_{\gamma},\epsilon_{0})}{\int}\underset{B(0,K)}{\int}\underset{\mathcal{R}_{j}}{\int}dz_{m}\wedge
d\bar{z}_{m}\cdots dz_{k}\wedge d\bar{z}_{k}\cdots dz_{\eta}\wedge
d\bar{z}_{\eta}dz_{\gamma}\wedge d\bar{z}_{\gamma}$
$\displaystyle=O(k^{-\frac{3}{q(1+\delta)}})$

Finally summing over all choices of ordering and of neighborhood, we get that

$\mu(\Omega_{3}(k))=O(k^{-\frac{3}{q(1+\delta)}})$

which is summable if $q<\frac{3}{1+\delta}$.

###  2.1 End of the proof

In order to have the required integrability, we should now adjust the
variables according to three conditions, namely:

  * •

$q<\frac{2}{Np}$

  * •

$q<2\frac{1+p}{1+\delta}$

  * •

$q<\frac{3}{1+\delta}$

So we fix $p$ and $\delta$ as

$\delta=\frac{1}{16N}<p=\frac{1}{8N}<\frac{1}{N}$

This yields the condition $q<2+C(N)$ with $C(N)$ some positive constant.
Overall choosing $q$ between $2$ and $2+C(N)$ we have shown the
$2+\kappa$-integrability of the Siegel-Veech transform for a $\kappa>0$.

######  Corollary.

For every $R>0$ the characteristic function of $B(0,R)$ is in
$L^{2+\kappa}(\mathcal{H},\mu)$.

###### Proof.

The proof is in [1] at theorem 3.3. We reproduce it here for completeness.

Cover the disk of radius $R$ with sectors of angle
$\frac{\epsilon_{0}^{2}}{R^{2}}$. By linearity is enough to show that for each
sector, the characteristic function $f$ has its Siegel-Veech transform
$\hat{f}\in L^{2+\kappa}(\mathcal{H},\mu)$.

Let $(g_{t})_{t\in\mathbb{R}}$ be the diagonal action and
$(r_{\theta})_{\theta\in\mathbb{R}}$ the rotational action of
$SL(2,\mathbb{R})$ on the strata of the moduli space. Moreover, let
$t_{0}=\log(\frac{R}{\epsilon_{0}})$ and $\theta_{0}$ be the center angle of
the sector. For any translation surface, if we apply
$g_{t_{0}}r_{-\theta_{0}}$, any saddle connection with period coordinate in
the sector will have in the new geometry a length less than $\epsilon_{0}$.
Calling $h$ the characteristic function of the disk of radius $\epsilon_{0}$,
since this two flow are measure preserving we have

$\int\hat{f}^{2+\kappa}d\mu<\int\hat{h}^{2+\kappa}d\mu<+\infty.$

∎

The proof of Theorem 1.4 is now an easy consequence of the previous corollary.

###### Proof of Theorem 1.4.

Indeed, take any $f\in B_{c}(\mathbb{R}^{2})$. By assumption there is a $R$
such that $|f|\leq\|f\|_{\infty}\chi_{B(0,R)}$, with $\chi_{B(0,R)}$ the
characteristic function on the ball $B(0,R)$. Positivity of the Siegel-Veech
transform yields the result. ∎

A corollary of Theorem 1.4 is that we can extend Theorem 1.6 to any ergodic
$SL(2,\mathbb{R})$-invariant measure $\mu$.

######  Theorem 2.5.

For any ergodic $SL(2,\mathbb{R})$-invariant measure $\mu$ and for any
$\phi\in C_{c}(\mathbb{C}^{2})$, for $\mu$-almost every
$\omega\in\mathcal{H}$, the circle average of $\hat{\phi}$ converge

$\underset{t\to\infty}{\lim}A_{t}\hat{\phi}(\omega)=\int_{\mathcal{H}}\hat{\phi}d\mu$

###### Proof.

The proof is identical to the proof of Proposition 2.2 of [2], but using the
that $\phi\in L^{1+\kappa}(\mathcal{H},\mu)$ with a general $\mu$. ∎

##  3 Bounds on the volume of some sets

To complete the discussion of the quadratic asymptotic for $N_{A}(\omega,R)$,
we need to revisit the proof of three technical lemmas, namely Lemma 5.2, 5.5
and 5.9 of [2].

The first one is not hard to extend thanks to the work of Dozier [5]. In our
setting, it states that if $\mu$ is a $SL(2,\mathbb{R})$-invariant measure
then:

######  Lemma 3.1.

For all $\epsilon,\kappa>0$, the $\mu$-volume of the set
$V_{1}(\epsilon,\kappa)\subset H$ of $\omega$ which have a saddle connection
of length at most $\epsilon$, and a non-homologous saddle connection with
length at most $\kappa$ is $O(\epsilon^{2}\kappa^{2})$

###### Proof.

The analog of this lemma for all $SL(2,\mathbb{R})$-invariant measure is
precisely the main result of Dozier, that is Theorem 1.1 of [5]. ∎

The second one bounds the integral of the Siegel-Veech transform of a function
with compact support on the thin part of the moduli space.

######  Lemma 3.2.

Let $N$ be the maximal number of edge in a Delaunay triangulation and choose
$\delta$ such that $\delta<1/2N$. Let $\mathcal{H}_{\epsilon}$ be the locus of
moduli space where the shortest saddle connection has length less than
$\epsilon_{1}$. Let $\psi$ be the Siegel-Veech transform of the characteristic
function of the ball $B(0,L_{0})\subset\mathbb{C}^{2}$ then

$\int_{\mathcal{H}_{\epsilon_{1}}}\psi d\mu=O(\epsilon_{1}^{1/N-2\delta}).$

To prove this lemma we recall a lemma of Athreya, Fairchild and Masur [2]
bounding

$N(\omega,R):=\Lambda_{\omega}\cap B(0,R)$

with respect to the length of the smallest saddle connection.

######  Lemma 3.3 (Lemma 5.4 of [2]).

For any $L_{0}>0$ and $\delta>0$, there exist $C(\delta,L_{0})$ such that for
any $L<L_{0}$ and any surface $(X,\omega)$ in the stratum we have

$N(\omega,L)\leq C\left(\frac{L}{l(\omega)}\right)^{1+\delta}.$

We should also control $N(\omega,R)$ with respect to the length of the
shortest saddle connection non-parallel to $\gamma$.

######  Lemma 3.4.

For $\omega\in\mathcal{H}$, if $\gamma$ is its shortest saddle connection and
$\gamma^{\prime}$ is the shortest saddle connection non-parallel to $\gamma$,
if $\gamma$ does not bound a cylinder or if $\gamma$ bound a cylinder of width
at least $\epsilon_{0}$, then

$N(\omega,\epsilon_{0})^{2}=O(l(\gamma^{\prime})^{-2N})$

###### Proof.

The proof has been done in Section 2 in the paragraphs
"The shortest non-parallel saddle connection is longer that a power of the
shortest and the shortest is not a cylinder curve"
and
"The shortest non-parallel saddle connection is longer that a power of the
shortest and the shortest is a cylinder curve and the height of the cylinder
is at least $\epsilon_{0}$". ∎

We can now adapt the proof of Lemma 5.5 of [2] to obtain our equivalent Lemma
3.2.

###### Proof of Lemma 3.2.

Following the proof of Athreya, Fairchild and Masur, we choose a real
$\sigma\in]0,1[$, and we define three families of set exhausting
$\mathcal{H}_{\epsilon}$.

We will use the notation of the proof of theorem 2.1 that is $\gamma$ is the
shortest saddle connection of a translation surface $\omega$ and $\epsilon$ is
the shortest saddle connection non-parallel with $\gamma$.

The first family is

$F(j)=:\\{\omega\in\mathcal{H},\sigma^{j+1}\leq|\gamma|\leq\sigma^{j},|\epsilon|\leq\sigma^{j/2N}\\}.$

In this case Lemma 3.2 say that

$\mu(F(j))=O(\sigma^{2j+2j/2N})$

and Lemma 3.3 that for $\omega\in F(j)$

$\psi(\omega)=O(\sigma^{-j(2+2\delta)}).$

This gives

$\int_{F(j)}\psi d\mu=O(\sigma^{j(1/N-2\delta)}).$

The second family is

$\displaystyle
G(j):=\\{\omega\in\mathcal{H},\sigma^{j+1}\leq|\gamma|\leq\sigma^{j},|\epsilon|\leq\sigma^{j/2N},\gamma\textrm{
is not on the boundary of a cylinder or}$ $\displaystyle\textrm{ is on the
boundary of a cylinder of height }\geq\epsilon_{0}\\}.$

In this case Lemma 3.1 gives that

$\mu(G(j))=O(\sigma^{2j})$

and Lemma 3.4 indicates that for $\omega\in G(j)$

$\psi(\omega)=O((\sigma^{j/2N})^{-2N})=O(\sigma^{-j}).$

And so

$\int_{G(j)}\psi d\mu=O(\sigma^{j}).$

Finally the third family is

$H(j):=\\{\omega\in\mathcal{H},\sigma^{j+1}\leq|\gamma|\leq\sigma^{j},\gamma\textrm{
is on the boundary of a cylinder of height }\leq\epsilon_{0}\\}.$

In this case Lemma 3.3 gives that for $\omega\in H(j)$

$\psi(\omega)=O(\sigma^{-j(2+2\delta)}).$

Moreover the measure of this set can be commputed as the measure of the set
$\Omega_{3}(k)$ is the last case of the proof of Theorem 2.1. Notice that in
the computation of the volume of $\Omega_{3}(k)$ the length of the shortest
non-parallel saddle connection does not interfere. So

$\mu(H(j))=O(\sigma^{3j})$

and

$\int_{H(j)}\psi d\mu=O(\sigma^{j(1-2\delta}).$

Finally choose $j_{0}$ so that
$\sigma^{j_{0}+1}\leq\epsilon_{1}\leq\sigma^{j_{0}}$ we have

$\int_{\mathcal{H}_{\epsilon_{1}}}\psi d\mu=O(\sum_{j\geq
j_{0}}\sigma^{j}+\sigma^{j(1-2\delta)}+\sigma^{j(1/N-2\delta)})=O(\sigma^{j_{0}(1/N-2\delta)})=O(\epsilon_{1}^{1/N-2\delta}).$

∎

The third one (Lemma 5.9 of [2]) bounds the measure of a set described by four
inequalities.

More precisely, given $L$, $\hat{\epsilon}$, $\epsilon^{\prime}$ and
$L^{\prime}\in\\{1/2,1\\}$ define
$\Omega(\hat{\epsilon},\epsilon^{\prime},L,L^{\prime})$ to be the set of
surfaces $\omega$ such that $\omega$ is $\hat{\epsilon}$-thick (that is has no
saddle connection of length less than $\hat{\epsilon}$), and there are
$(z,w)\in\left(\Lambda_{\omega}\cap B(0,L)\right)^{2}$, where at least one of
the following holds:

  * •

$1-\epsilon^{\prime}\leq\frac{|\operatorname{Im}(w)|}{|\operatorname{Im}(z)|}\leq
1+\epsilon^{\prime}$

  * •

$(1-\epsilon^{\prime})A\leq|z\wedge w|\leq(1+\epsilon^{\prime})A$

  * •

$|\operatorname{Im}(z)-L^{\prime}|<\epsilon^{\prime}$

  * •

$(1-\epsilon^{\prime})\operatorname{Im}(z)\leq|\operatorname{Re}(z)|\leq(1+\epsilon^{\prime})\operatorname{Im}(z)$

######  Lemma 3.5.

There exists $C$ and $D>0$ so that for all $\hat{\epsilon},\epsilon^{\prime}$

$\mu(\Omega(\hat{\epsilon},\epsilon^{\prime},L,L^{\prime}))\leq
C\frac{\epsilon^{\prime}}{\hat{\epsilon}^{D}}$

###### Proof.

As the $\hat{\epsilon}$-thick part of the moduli space is compact and the
affine manifold $\mathcal{M}$ is closed, their intersection is also compact.
Each point has a neighborhood in which there is a triangulation that is a
Delaunay triangulation for every point in this neighborhood. We fix a finite
cover of this family of neighborhood. Then taking one of these neighborhood,
we will work with a fixed Delaunay triangulation.

Let’s take a maximal family of $\mathcal{M}$-independent saddle connections
which are edges of the Delaunay triangulation $t_{1},t_{2},\cdots,t_{k}$ so
that the $\mu$ measure is the Lebesgue measure under their period coordinates.
We decompose the period coordinates as $t_{j}=x_{j}+iy_{j}$.

Then we complete this family with $t_{k+1},\cdots,t_{n}$ to have a basis of
the $H_{1}(X,\Sigma,\mathbb{C})$.

The length of any $t_{i}$ is bounded below by $\hat{\epsilon}$ and by the
Paragraph 4 of [12] also bound above by $1/\hat{\epsilon}$.

Then if two saddle connections $z,w$ have length less than $L$ and satisfies
the first inequality, we can decompose $z=\sum_{i=1}^{n}z_{i}t_{i}$ and
$w=\sum_{i=1}^{n}w_{i}t_{i}$ in the homology, with integer coefficient.

The coefficients are $O(\frac{1}{\hat{\epsilon}})$ since the Delaunay
triangulation is efficient.

Since every $t_{i}$, for $k<i\leq n$ are linear combinations of the $t_{j}$
for $1\leq j\leq k$, we can write $z=\sum_{i=1}^{k}z_{i}^{\prime}t_{i}$ and
$w=\sum_{i=1}^{k}w_{i}^{\prime}t_{i}$ with coefficients which are still
$O(\frac{1}{\hat{\epsilon}})$. We fix a pair of tuple of coefficients
$(z_{i}^{\prime},w_{i}^{\prime})$ with $z_{i}^{\prime}\neq w_{i}^{\prime}$.

We are interested in computing the Lebesgue volume of vectors $s_{i}$ of
lengths less than $L$ such that

$\left|\frac{\sum_{i=1}^{k}z_{i}^{\prime}y_{i}}{\sum_{i=1}^{k}w_{i}^{\prime}y_{i}}-1\right|\leq\epsilon^{\prime}$

This volume is smaller than the volume of the set of vectors of lengths less
than $L$ such that

$\left|\sum_{i=1}^{k}(z_{i}^{\prime}-w_{i}^{\prime})y_{i}\right|\leq\epsilon^{\prime}O(\frac{1}{\hat{\epsilon}^{2}})$

This is the intersection of the sphere of radius $L$ centered at $0$ and the
$O(\frac{\epsilon^{\prime}}{\hat{\epsilon}^{2}})$ neighborhood of a hyperplane
in a $k$ dimensional coordinate vector space, whose volume is
$O(\frac{\epsilon^{\prime}}{\hat{\epsilon}^{2}})$. The number of choices of
coefficient is $O(\frac{1}{\hat{\epsilon}^{k}})$. By summing the estimates for
each choice, we get the bound of the first point.

The other points lead to same the kind of inequalities and are demonstrated
similarly.

∎

At this point we have shown that given $\mu$ an ergodic
$SL(2,\mathbb{R})$-invariant measure, there is a $c_{\mu}(A)$ such that for
$\mu$-almost every $\omega\in\mathcal{H}$

$\underset{R\to\infty}{\lim}\frac{N_{A}(\omega,R)}{R^{2}}=c_{\mu}(A).$

##  4 Effectivization of the counting

In order to get the power saving error term of Theorem 1.2, we mainly need to
control two quantities. The first one is the difference between the circle
average and the integral of the Siegel-Veech transform of $h_{A}$. To do this
we will use an effective version of Nevo ergodic theorem 1.5. The second
quantity is the sum of the four error terms $e_{t}^{i}$.

###  4.1 Effective Nevo ergodic theorem

We define now the class of function, we should work with.

######  Definition 4.1.

A function $f\in L^{2}(\mathcal{H},\mu)$ is called K-$smooth$ of degree one if

$\pi_{\mathcal{H}}(\theta)f:=\underset{\theta\to
0}{\lim}\frac{1}{\theta}(r_{\theta}^{*}f-f)$

exists with respect to the $L^{2}(\mathcal{H},\mu)$-norm.

We define the _Sobolev norm_ by

$S_{K}(f)^{2}=\|f\|^{2}_{2}+\|\pi_{\mathcal{H}}(\theta)f\|_{2}^{2}$

The $K$_-Sobolev space_ is then define as

$\Sigma_{K}(\mathcal{H})=\\{f\in L^{2}(\mathcal{H},\mu),S_{K}(f)<+\infty\\}.$

For a function $g$ from $\mathbb{C}^{2}$ to $\mathbb{R}$ we define

$\partial_{\theta}g:=\underset{\theta\to
0}{\lim}\frac{r_{\theta}^{*}g-g}{\theta}.$

Note that $(\partial_{\theta}g)^{SV}=\pi_{\mathcal{H}}(\theta)\hat{g}$.

We can bound the Siegel-Veech transform of function with compact support and
its $S_{K}$-norm of with data from the original function.

######  Lemma 4.1.

Given $\alpha_{1}>2$, let $g:\mathbb{C}^{2}\to\mathbb{R}$ be function with
compact support and let $\epsilon>0$. We have that

$|\hat{g}(\omega)|\leq C_{supp(g)}l(\omega)^{-\alpha_{1}}\|g\|_{\infty}$

and if $g$ is differentiable,

$S_{K}\left(\hat{g}\chi_{l(\cdot)\geq\epsilon}-\int_{\mathcal{H}}\hat{g}\chi_{l(\cdot)\geq\epsilon}d\mu\right)^{2}=O(\frac{\|g\|_{\infty}^{2}+\|\partial_{\theta}g\|_{\infty}^{2}}{\epsilon^{2\alpha_{1}}})$

where the implied constant depend on the support.

###### Proof.

For the first point, taking any surface $\omega\in\mathcal{H}$, if a pair of
saddle connection are in $supp(g)$, then they both belong to a compact set
$B\subset\mathbb{R}^{2}$. Hence using Lemma 3.3, we got

$|\hat{g}(\omega)|\leq\|g\|_{\infty}(\\#\Lambda(\omega)\cap B)^{2}\leq
C_{supp(g)}l(\omega)^{-\alpha_{1}}\|g\|_{\infty}$

For the second point

$\displaystyle\|\hat{g}\chi_{l(\cdot)\geq\epsilon}-\int_{\mathcal{H}}\hat{g}\chi_{l(\cdot)\geq\epsilon}d\mu\|_{2}^{2}$
$\displaystyle\leq\int_{l(\cdot)\geq\epsilon}\hat{g}^{2}d\mu+\left(\int_{\mathcal{H}}\hat{g}\chi_{l(\cdot)\geq\epsilon}d\mu\right)^{2}$
$\displaystyle\leq
O(\int_{l(\cdot)\geq\epsilon}l(\omega)^{-2\alpha_{1}}\|g\|_{\infty}^{2}d\mu)+O(\left(\int_{l(\cdot)\geq\epsilon}l(\omega)^{-\alpha_{1}}\|g\|_{\infty}d\mu\right)^{2})$
$\displaystyle=O(\|g\|_{\infty}^{2}\epsilon^{-2\alpha_{1}})$

Where the second inequality comes from the first point of this lemma.

Then note that $\pi_{\mathcal{H}}(\theta)\chi_{l(\cdot)\geq\epsilon}=0$ and
$\pi_{\mathcal{H}}(\theta)\hat{g}=(\partial_{\theta}g)^{SV}$ and so

$\pi_{\mathcal{H}}(\theta)\hat{g}\chi_{l(\cdot)\geq\epsilon}=(\partial_{\theta}g)^{SV}\chi_{l(\cdot)\geq\epsilon}.$

So, in the same flavor as the previous estimate

$\displaystyle\|\pi_{\mathcal{H}}(\theta)\hat{g}\chi_{l(\cdot)\geq\epsilon}\|_{2}^{2}$
$\displaystyle=\int_{l(\cdot)\geq\epsilon}\left(\partial_{\theta}g^{SV}\right)^{2}d\mu$
$\displaystyle\leq\int_{l(\cdot)\geq\epsilon}l(\omega)^{-2\alpha_{1}}\|\partial_{\theta}g\|_{\infty}^{2}d\mu$
$\displaystyle=O(\epsilon^{-2\alpha_{1}}\|\partial_{\theta}g\|_{\infty}^{2})$

This end the demonstration of this lemma.

∎

The following statement is an effective version of 1.5 which is included in
Theorem 3.5 of [14].

######  Theorem 4.2.

There is a $\lambda^{\prime}>0$ and a $C>0$ such that for all $t>1$ and any
$f\in\Sigma_{K}(\mathcal{H})$, we have

$\|A_{t}f-\int_{\mathcal{H}}fd\mu\|^{2}_{2}\leq
Ce^{-2\lambda^{\prime}t}S_{K}(f)^{2}.$

Furthermore, if $(t_{n})\subset{\mathbb{R}_{+}}$ and $\eta_{1}$ satisfy

$\sum_{n\in\mathbb{N}}e^{-\lambda^{\prime}\eta_{1}t_{n}}<\infty$ (4)

then for almost all $\omega\in\mathcal{H}$ there is $n_{0}=n_{0}(\omega)$ such
that for all $n\geq n_{0}$

$|A_{t_{n}}f(\omega)-\int_{\mathcal{H}}fd\mu|\leq
Ce^{-2(\eta-\frac{\eta_{1}}{2})\lambda^{\prime}t_{n}}S_{K}(f)^{2}$

with $\eta=\frac{1}{\lambda^{\prime}+1}$

From now on, we fix a sequence $(t_{n})$ and an $\eta_{1}$ satisfying equation
4. We will call $\lambda=\frac{\eta-\eta_{1}/2}{\lambda^{\prime}+1}$.

###  4.2 Circle average convergence

The aim of this subsection is to estimate

$\left|A_{t}\hat{h}_{A}(\omega)-\int_{\mathcal{H}}\hat{h}_{A}d\mu\right|$

To begin, we will need a bounding lemma (Lemma 4.4 below) which is based on
the following theorem.

######  Theorem 4.3 (See [6] Thm 5.2 and Lem 5.5 and [16] Cor 2.8).

For any $\omega\in\mathcal{H}$, and for any $1\leq\alpha<2$,

$\underset{t>0}{\sup}A_{t}(l^{-\alpha})(\omega)<\infty.$

The bound can be taken uniform as $\omega$ ranges over compact sets in
$\mathcal{H}$. Moreover,

$l(\cdot)^{-\alpha}\in L^{1}(\mathcal{H},\mu).$

Using this theorem we can bound the circle average of a Siegel-Veech transform
of a function with compact support on the thin part.

######  Lemma 4.4.

There is an $\alpha_{2}>0$ such that for any $g:\mathbb{C}^{2}\to\mathbb{R}$
be a positive function with compact support, any $\epsilon>0$, any
$\omega\in\mathcal{H}$ there is a constant $K(\omega,g)$ with

$|A_{t_{n}}(\hat{g}\chi_{l(\cdot)\leq\epsilon})|\leq\epsilon^{\alpha_{2}}K(\omega,g).$

Moreover if $h\leq g$ then $K(\omega,h)\leq K(\omega,g)$.

###### Proof.

Denote by $\chi_{\epsilon}:=\chi_{l(\cdot)\leq\epsilon}$ and observe that, by
taking $\alpha_{2}$ such that $\alpha_{2}\kappa<2(1+\kappa)$, one has

$\displaystyle|A_{t}(\hat{g}\chi_{\epsilon})(\omega)|$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\hat{g}(g_{t}r_{\theta}\omega)\chi_{\epsilon}(g_{t}r_{\theta}\omega)d\theta$
$\displaystyle\leq\frac{\epsilon^{\alpha_{2}}}{2\pi}\int_{0}^{2\pi}\hat{g}(g_{t}r_{\theta}\omega)l^{-\alpha_{2}}(g_{t}r_{\theta}\omega)\chi_{\epsilon}(g_{t}r_{\theta}\omega)d\theta$
$\displaystyle\leq\frac{\epsilon^{\alpha_{2}}}{2\pi}\left(\int_{0}^{2\pi}l^{\frac{\alpha_{2}\kappa}{{}^{1+\kappa}}}(g_{t}r_{\theta}\omega)d\theta\right)^{\frac{\kappa+1}{\kappa}}\left(\int_{0}^{2\pi}\hat{g}(g_{t}r_{\theta}\omega)^{1+\kappa}\chi_{\epsilon}(g_{t}r_{\theta}\omega)^{1+\kappa}d\theta\right)^{\frac{1}{1+\kappa}}$

where the second inequality is the Hölder inequality. Then by Theorem 4.3 and
the choice of $\alpha_{2}$ the first integral of the last line is finite. The
second integral, with Theorem 2.5, converges to

$\int_{\mathcal{H}}\hat{g}^{1+\kappa}\chi_{\epsilon}^{1+\kappa}d\mu.$

So the second integral is bound for all $t$ by a quantity called
$K^{\prime}(\omega,g)$. This completes the argument.

The last affirmation is obvious looking at what defined the constants. ∎

As $\hat{h}_{A}$ does not satisfy the hypothesis of Theorem 4.2, we will
approach $\hat{h}_{A}$ by a family of function depending on two parameters.
The first one will smooth $h_{A}$ and the second will restrict its Siegel-
Veech transform to the thick part of the moduli space.

######  Definition 4.2.

Let $\eta:\mathbb{C}^{2}\to[0,1]$ be a smooth function with support in
$B(0,2)$ and equal to $1$ on $B(0,1)$ and such that

$\int_{\mathbb{C}^{2}}\eta dLeb=1.$

Let

$\eta_{s}(y)=\frac{1}{s^{4}}\eta(\frac{y}{s})$

and

$R_{A,s}(\mathcal{T}):=\\{x\in\mathbb{C}^{2},dist(x,R_{A}(\mathcal{T}))\leq
s\\}$

We consider

$g_{A,s}(x)=(\chi_{R_{A,s}(\mathcal{T})}*\eta_{s})(x)=\int_{\mathbb{C}^{2}}\eta_{s}(x-y)\chi_{R_{A,s}(\mathcal{T})}(y)dLeb(y)$

$g_{A,s}$ is a function from $\mathbb{C}^{2}$ to $[0,1]$ which is
differentiable, with a bounded differential and compact support. An easy
consequence is that the $K$-derivative, $\partial_{\theta}g_{A,s}$ is also
bound and is $O(\frac{1}{s})$.

The supports of the function $g^{t}_{A}$ are increasing meaning that if
$s<t<1$ then

$supp(g_{s,A})\subset supp(g_{t,A})$

In particular $supp(g_{t,A})\subset supp(g_{1,A})$ for every $t\leq 1$. We
bound every function $g_{t,A}$ by the characteristic function of
$supp(g_{1,A})$.

So using the first part of Lemma 4.1 we have that

$|\hat{g}_{A,s}(\omega)|\leq Cl(\omega)^{-\alpha_{1}}$

with no dependency on $s$ for the constant.

This family of function also satisfy the following estimates.

######  Lemma 4.5.

There are $\kappa_{1},\kappa_{2},\kappa_{3}>0$ such that, for almost every
$\omega\in\mathcal{H}$ there is a $n_{0}$ such that for $n\geq n_{0}(\omega)$
we have:

  * •

$\left|A_{t_{n}}\hat{h}_{A}(\omega)-A_{t_{n}}\hat{g}_{A,s}(\omega)\right|\leq
O_{\omega}(s^{\kappa_{1}})+O(s^{-\kappa_{2}})e^{-2\lambda t_{n}}$

  * •

$\left|\int_{\mathcal{H}}\hat{g}_{A,s}d\mu-\int_{\mathcal{H}}\hat{h}_{A}d\mu\right|\leq
O(s^{\kappa_{3}})$

###### Proof.

Let’s take

$F_{A,s}(\mathcal{T}):=\\{x\in\mathbb{C}^{2},dist(x,\partial
R_{A}(\mathcal{T}))\leq s\\}$

We consider

$\psi_{s}(x)=(\chi_{F_{A,s}(\mathcal{T})}*\eta_{s})(x)$

We have that $0\leq g_{A,s}-h_{A}\leq\psi_{s}$, $\psi_{s}\leq 1$, the support
of $\psi_{s}$ is compact, $\psi_{s}$ is differentiable with bounded
differential. So,calling $\chi_{\epsilon}$ the characteristic function of the
$\epsilon$-thin part, we can say using Theorem 4.2 that for almost every
$\omega\in\mathcal{H}$ and $n\geq n_{0}(\omega)$

$\left|A_{t_{n}}\hat{h}_{A}(\omega)-A_{t_{n}}\hat{g}_{A,s}(\omega)\right|\leq\left|A_{t_{n}}\hat{\psi}_{s}(\omega)\chi_{\epsilon}\right|+\left|A_{t_{n}}\hat{\psi}_{s}(\omega)(1-\chi_{\epsilon})\right|$

As $\psi_{s}\leq\psi_{1}$, by Lemma 4.4 we already have that

$\left|A_{t_{n}}\hat{\psi}_{s}(\omega)\chi_{\epsilon}\right|=O(\epsilon^{\alpha_{2}})K(\omega,\psi_{s})\leq
O(\epsilon^{\alpha_{2}})K(\omega,\psi_{1})$

And using Theorem 4.2,

$\left|A_{t_{n}}\hat{\psi}_{s}(\omega)(1-\chi_{\epsilon})\right|\leq\left|\int_{l(\cdot)\geq\epsilon}\hat{\psi}_{s}d\mu\right|+CS_{K}\left(\hat{\psi}_{s}(1-\chi_{\epsilon})-\int_{\mathcal{H}}\hat{\psi}_{s}(1-\chi_{\epsilon})d\mu\right)^{2}e^{-2\lambda
t_{n}}.$

Using the second part of Lemma 4.1, we have easily that

$S_{K}\left(\hat{\psi}_{s}(1-\chi_{\epsilon})-\int_{\mathcal{H}}\hat{\psi}_{s}(1-\chi_{\epsilon})d\mu\right)^{2}=O(\frac{1}{\epsilon^{2\alpha_{1}}s^{2}}).$

Moreover

$\left|\int_{\mathcal{H}}\hat{g}_{A,s}d\mu-\int_{\mathcal{H}}\hat{h}_{A}d\mu\right|\leq\int_{\mathcal{H}}\hat{\psi}_{s}d\mu=\int_{l\leq\epsilon}\hat{\psi}_{s}d\mu+\int_{l\geq\epsilon}\hat{\psi}_{s}d\mu.$

For the first integral by Lemma 3.2, if $N$ is the maximal number of edge in a
Delaunay triangulation and $\delta$ satisfies $\delta<\frac{1}{2N}$ then

$\int_{l\leq\epsilon}\hat{\psi}_{s}d\mu=O(\epsilon^{1/N-2\delta}).$

Then by Lemma 4.1, for each $\omega$ in the $\epsilon$ thick part,
$|\hat{\psi}_{s}(\omega)|=O_{supp(\psi_{s})}(\frac{1}{\epsilon^{\alpha_{1}}})=O_{supp(\psi_{1})}(\frac{1}{\epsilon^{\alpha_{1}}})$.
As the support of $\psi_{s}$ is included in
$\Omega(\epsilon_{1},s,L,L^{\prime})$ for a $L$ big enough, Lemma 3.5 say that
the measure of the support of $\hat{\psi}_{s}$ is
$O(\frac{s}{\epsilon_{1}^{D}})$. Putting everything together we have that

$\int_{\mathcal{H}}\hat{\psi}_{s}d\mu\leq
O(\epsilon^{1/N-2\delta})+O(\frac{s}{\epsilon^{D+\alpha_{1}}})$

Taking $\epsilon=s^{\frac{1}{D+\alpha_{1}+1/N-2\delta}}$ we got that

$\int_{\mathcal{H}}\hat{\psi}_{s}d\mu\leq
O(s^{\frac{1/N-2\delta}{D+\alpha_{1}+1/N-2\delta}}).$

This concludes the second point.

For the first point we have

$\left|A_{t_{n}}\hat{h}_{A}(\omega)-A_{t_{n}}\hat{g}_{A,s}(\omega)\right|\leq
O(\epsilon^{\alpha_{2}})K(\omega)+O(\frac{s}{\epsilon^{D+\alpha_{1}}})+O(\frac{1}{\epsilon^{2\alpha_{1}}s^{2}})e^{-2\lambda
t_{n}}$

Taking here $\epsilon=s^{\frac{1}{D+2\alpha_{1}+\alpha_{2}}}$ we have

$\left|A_{t_{n}}\hat{h}_{A}(\omega)-A_{t_{n}}\hat{g}_{A,s}(\omega)\right|\leq
O_{\omega}(s^{\frac{\alpha_{2}}{D+2\alpha_{1}+\alpha_{2}}})+O(\frac{1}{s^{\frac{\alpha_{1}}{D+2\alpha_{1}+\alpha_{2}}+2}})e^{-2\lambda
t_{n}}$

∎

We now fix an $\epsilon>0$ and call $\chi_{\epsilon}$ the characteristic
function of $l^{-1}(]0,\epsilon[)$ on $\mathcal{H}$. We have that,

$\displaystyle\left|A_{t}\hat{h}_{A}(\omega)-\int_{\mathcal{H}}\hat{h}_{A}d\mu\right|$
$\displaystyle\leq\left|A_{t}\hat{h}_{A}(\omega)-A_{t}\hat{g}_{A,s}(\omega)\right|+\left|A_{t}\hat{g}_{A,s}\chi_{\epsilon}(\omega)\right|$
$\displaystyle+\left|A_{t}\hat{g}_{A,s}(1-\chi_{\epsilon})(\omega)-\int_{\mathcal{H}}\hat{g}_{A,s}(1-\chi_{\epsilon})d\mu\right|$
(5)
$\displaystyle+\left|\int_{\mathcal{H}}\hat{g}_{A,s}\chi_{\epsilon}d\mu\right|+\left|\int_{\mathcal{H}}\hat{g}_{A,s}d\mu-\int_{\mathcal{H}}\hat{h}_{A}d\mu\right|$

In the previous inequality we approximated $h_{A}$ by $g_{A,s}$ and then
treated separately the thin and thick part of the moduli space. We can bound
these terms with the following lemma.

######  Lemma 4.6.

There are $\alpha_{3},C>0$ such that

  * •

$S_{K}(\hat{g}_{A,s}(1-\chi_{\epsilon})-\int_{\mathcal{H}}\hat{g}_{A,s}(1-\chi_{\epsilon})d\mu)^{2}\leq\epsilon^{-\alpha_{1}}O(\frac{1}{s^{2}})$

  * •

$\int_{\mathcal{H}}\hat{g}_{A,s}\chi_{\epsilon}d\mu\leq
C_{3}\epsilon^{\alpha_{3}}$

  * •

$|A_{t}(\hat{g}_{A,s}\chi_{\epsilon})(\omega)|\leq\epsilon^{\alpha_{2}}K(\omega)$

###### Proof.

The first point is just Lemma 4.1.

For the second point, choosing $\alpha_{3}$ such that
$\alpha_{3}+\alpha_{1}<2$, with Theorem 4.3 we have

$\displaystyle\int_{\mathcal{H}}\hat{g}_{A,s}\chi_{\epsilon}d\mu$
$\displaystyle\leq\int_{l(\omega\leq\epsilon)}\hat{g}_{A,s}d\mu$
$\displaystyle\leq\int_{l(\omega\leq\epsilon)}Cl(\omega)^{-\alpha_{1}}d\mu$
$\displaystyle\leq\int_{l(\omega\leq\epsilon)}C\frac{\epsilon^{\alpha_{3}}}{l(\omega)^{\alpha_{3}}}l(\omega)^{-\alpha_{1}}d\mu$
$\displaystyle\leq\epsilon^{\alpha_{3}}C\|l^{-\alpha_{3}-\alpha_{1}}\|_{1}$

The third point is an application of Lemma 4.4, in which we bound $g_{A,s}$ by
$g_{A,1}$ to get rid of the dependency on $s$. ∎

Going back to inequality 4.2, we have that for almost every
$\omega\in\mathcal{H}$ and $n\geq n_{0}(\omega)$

$\displaystyle\left|A_{t_{n}}\hat{h}_{A}(\omega)-\int_{\mathcal{H}}\hat{h}_{A}d\mu\right|$
$\displaystyle\leq\left|A_{t_{n}}\hat{h}_{A}(\omega)-A_{t_{n}}\hat{g}_{A,s}(\omega)\right|+\left|A_{t_{n}}\hat{g}_{A,s}\chi_{\epsilon}(\omega)\right|$
$\displaystyle+\left|A_{t_{n}}\hat{g}_{A,s}(1-\chi_{\epsilon})(\omega)-\int_{\mathcal{H}}\hat{g}_{A,s}(1-\chi_{\epsilon})d\mu\right|$
$\displaystyle+\left|\int_{\mathcal{H}}\hat{g}_{A,s}\chi_{\epsilon}d\mu\right|+\left|\int_{\mathcal{H}}\hat{g}_{A,s}d\mu-\int_{\mathcal{H}}\hat{h}_{A}d\mu\right|$
$\displaystyle\leq O_{\omega}(s^{\kappa_{1}})+O(s^{-\kappa_{2}})e^{-2\lambda
t_{n}}+\epsilon^{\alpha_{2}}K(\omega)$ $\displaystyle+Ce^{-2\lambda
t_{n}}S_{K}(\hat{g}_{A,s}(1-\chi_{\epsilon})-\int_{\mathcal{H}}\hat{g}_{A,s}(1-\chi_{\epsilon})d\mu)^{2}$
$\displaystyle+C_{3}\epsilon^{\alpha_{3}}+O(s^{\kappa_{3}})$
$\displaystyle\leq
e^{-2\lambda{t_{n}}}O(s^{-\kappa_{2}}+\frac{1}{s^{2}\epsilon_{1}})+O_{\omega}(s^{\kappa_{1}}+s^{\kappa_{3}}+\epsilon^{\alpha
2}+\epsilon^{\alpha_{1}})$

Putting $\epsilon=e^{-f_{1}t_{n}}$ and $s=e^{-f_{2}t_{n}}$ we have

$\left|A_{t_{n}}\hat{h}_{A}(\omega)-\int_{\mathcal{H}}\hat{h}_{A}\right|\leq
O_{\omega}(e^{-t_{n}(2\lambda-\kappa_{2}f_{2})}+e^{-t_{n}(2\lambda-2f_{2}-\alpha_{1}f_{1})}+e^{-f_{2}\kappa_{1}t_{n}}+e^{-f_{2}\kappa_{3}t_{n}}+e^{-f_{1}\alpha_{2}t_{n}}+e^{-f_{1}\alpha_{3}t_{n}})$

Choosing wisely $f_{1}$ and $f_{2}$ we have for a $f>0$

$\left|A_{t_{n}}\hat{h}_{A}(\omega)-\int_{\mathcal{H}}\hat{h}_{A}\right|\leq
O_{\omega}(e^{-t_{n}f}).$

###  4.3 Error terms

Equation (4.10) of [2] indicates that

$\left|N_{A}^{*}(\omega,e^{t})-\pi
e^{2t}(A_{t}\hat{h}_{A})(\omega)\right|\leq|m_{t}(\omega)|+\sum_{i=1}^{4}|e^{i}_{t}(\omega)|$

where the different terms are explained above.

Equation (5.4) of [2] gives that $|m_{t}(\omega)|=O(e^{-2t})$. We will
concentrate on the four other terms.

For the other error terms we choose $\hat{\epsilon}>0$ to work separately on
the thin and thick parts of the moduli space.

If $\omega$ is on the $\hat{\epsilon}$ thin part of the moduli space,
according to Corollary 5.8 of [2], we have that

$|E^{k}_{t}\cap\Lambda_{\omega}^{2}|=O(\hat{\epsilon}^{1/N-2\delta}e^{2t}).$

We should now pick $\epsilon^{\prime}$ satisfying every inequality in the
proof of theorem 5.1 in [2] which are on the beginning of each paragraph
"Error term $E^{i}_{t}$".

Taking larger bounds than these inequalities, we can find a $K$ such that
$\epsilon^{\prime}$ satisfies all the conditions if

$Ke^{-2t}\leq\epsilon^{\prime}.$

To have estimates on the error terms on the thick part, it is needed to bound
$|A_{t}h|$ where $h$ is the characteristic function of
$\Omega(\frac{\hat{\epsilon}}{\sqrt{8}\pi},\epsilon^{\prime},L,L^{\prime})$.
As $h$ is not $K$-smooth we will also approach it by a family of function
defined below.

To do that let’s call $\Theta(\hat{\epsilon},\epsilon^{\prime},L,L^{\prime})$
the set of pair $(w,z)$ in $\mathbb{C}^{2}$ such that

$\hat{\epsilon}\leq\min(|w|,|z|)\leq\max(|w|,|z|)\leq L$

and which verified at least one of these equations

  * $\bullet$

$1-\epsilon^{\prime}\leq\frac{|\operatorname{Im}(w)|}{|\operatorname{Im}(z)|}\leq
1+\epsilon^{\prime}$

  * $\bullet$

$(1-\epsilon^{\prime})A\leq|z\wedge w|\leq(1+\epsilon^{\prime})A$

  * $\bullet$

$|\operatorname{Im}(z)-L^{\prime}|<\epsilon^{\prime}$

  * $\bullet$

$(1-\epsilon^{\prime})\operatorname{Im}(z)\leq|\operatorname{Re}(z)|\leq(1+\epsilon^{\prime})\operatorname{Im}(z)$

We consider $\eta_{s}$ the family of bump functions defined in the previous
part and call
$\xi_{s}=\eta_{s}*\chi_{\\{(z,w),dist((z,w),\Theta(\frac{\hat{\epsilon}}{\sqrt{8}\pi},\epsilon^{\prime},L,L^{\prime}))\leq
s\\}}$.

$\xi_{s}$ have the following properties

  1. 1.

$\xi_{s}(z,w)\in[0,1]$

  2. 2.

$\xi_{s}\geq\chi_{\Theta(\frac{\hat{\epsilon}}{\sqrt{8}\pi},\epsilon^{\prime},L,L^{\prime})}$

  3. 3.

$supp(\xi_{s})\subset\Theta(\frac{\hat{\epsilon}}{\sqrt{8}\pi}-s,\epsilon^{\prime}+\frac{s}{\hat{\epsilon}},L+s,L^{\prime})\subset\Theta(\frac{\hat{\epsilon}}{2\sqrt{8}\pi},\epsilon^{\prime}+\frac{s}{\hat{\epsilon}},2L,L^{\prime})$
for $s$ small enough

  4. 4.

$\xi_{s}$ is differentiable with its norm bound by $O(1/s)$.

Then $h$ is bound by the Siegel-Veech transform of these functions time the
characteristic function of the $\frac{\hat{\epsilon}}{\sqrt{8}\pi}$-thick part
of the moduli space

$\chi_{l(\cdot\geq\frac{\hat{\epsilon}}{\sqrt{8}\pi})}\hat{\xi}_{s}:=\Xi_{s,\hat{\epsilon}}.$

$\Xi_{s,\hat{\epsilon}}$ has its support on
$\Omega(\frac{\hat{\epsilon}}{\sqrt{8}\pi},\epsilon^{\prime}+\frac{\sqrt{8}\pi
s}{\hat{\epsilon}},2L,L^{\prime})$ and is $K$-smooth.

Using the first part of Lemma 4.1 we have that for all $\omega\in\mathcal{H}$

$|\Xi_{s,\hat{\epsilon}}(\omega)|\leq O(\hat{\epsilon}^{-\alpha_{1}}).$

Using the second part of Lemma 4.1 we have that
$S_{K}(\Xi_{s,\hat{\epsilon}}-\int_{\mathcal{H}}\Xi_{s,\hat{\epsilon}}d\mu)^{2}=O(\frac{1}{\hat{\epsilon}^{2\alpha_{1}}s^{2}})$

Moreover using Lemma 3.5 we got

$\int_{\mathcal{H}}\Xi_{s,\hat{\epsilon}}d\mu\leq\|\Xi_{s,\hat{\epsilon}}\|_{\infty}\mu(\Omega(\frac{\hat{\epsilon}}{\sqrt{8}\pi},\epsilon^{\prime}+\frac{s}{\hat{\epsilon}},2L,L^{\prime}))=O(\hat{\epsilon}^{-\alpha_{1}})O(\frac{\epsilon^{\prime}+\frac{s}{\hat{\epsilon}}}{\hat{\epsilon}^{D}}).$

We have that for $\mu$-almost every $\omega\in\mathcal{H}$ that for $n\geq
n_{0}(\omega)$

$|A_{t_{n}}\hat{h}|\leq|A_{t_{n}}\Xi_{s,\hat{\epsilon}}|\leq\int_{\mathcal{H}}\Xi_{s,\hat{\epsilon}}d\mu+CS_{K}(\Xi_{s,\hat{\epsilon}}-\int_{\mathcal{H}}\Xi_{s,\hat{\epsilon}}d\mu)^{2}e^{-\lambda
t_{n}}=O(\frac{\epsilon^{\prime}+\frac{s}{\hat{\epsilon}}}{\hat{\epsilon}^{D+\alpha_{1}}}+\frac{e^{-\lambda
t_{n}}}{\hat{\epsilon}^{2\alpha_{1}}s^{2}}).$

Taking $s=\epsilon^{\prime}\hat{\epsilon}$ we have

$|A_{t_{n}}\hat{h}|=O(\frac{\epsilon^{\prime}}{\hat{\epsilon}^{D+\alpha_{1}}}+\frac{e^{-\lambda
t_{n}}}{\epsilon^{\prime 2}\hat{\epsilon}^{2\alpha_{1}+2}})$

Following the rest of the proof of Proposition 5.1 of Athreya, Fairchild and
Masur in [2], on the thick part of the moduli space,
$|E^{k}_{t_{n}}\cap\Lambda_{\omega}^{2}|$ is bound by the product of two
quantities.

The first one is a counting of acceptable sector $\\#I(\theta_{i})$ which is
shown to be $O(e^{2t}|A_{t}\hat{h}|)$. The second one is the maximum number of
pair of saddle connections in each sector which they showed to be
$O(\frac{1}{\hat{\epsilon}^{1+\delta}})$.

With the previous computation we have that

$\\#I(\theta_{i})=e^{2t_{n}}|A_{t_{n}}\hat{h}|=e^{2t_{n}}O(\frac{\epsilon^{\prime}}{\hat{\epsilon}^{D+\alpha_{1}}}+\frac{e^{-\lambda
t_{n}}}{\epsilon^{\prime 2}\hat{\epsilon}^{2\alpha_{1}+2}})$

and then in the thick part

$|E^{k}_{t_{n}}\cap\Lambda_{\omega}^{2}|=e^{2t_{n}}O\left((\frac{1}{\hat{\epsilon}})^{1+\delta}\left(\frac{\epsilon^{\prime}}{\hat{\epsilon}^{D+\alpha_{1}}}+\frac{e^{-\lambda
t_{n}}}{\epsilon^{\prime 2}\hat{\epsilon}^{2\alpha_{1}+2}}\right)\right).$

Grouping the thin and thick part we have

$|E^{k}_{t_{n}}\cap\Lambda_{\omega}^{2}|=e^{2t_{n}}O\left(\hat{\epsilon}^{1/N-2\delta}+(\frac{1}{\hat{\epsilon}})^{1+\delta}\left(\frac{\epsilon^{\prime}}{\hat{\epsilon}^{D+\alpha_{1}}}+\frac{e^{-\lambda
t_{n}}}{\epsilon^{\prime 2}\hat{\epsilon}^{2\alpha_{1}+2}}\right)\right).$

Then by Lemma 4.1 of [2], there is a $T_{0}$ such that for $t\geq T_{0}$,

$A_{t}(h_{A})(z,w)\pi e^{2t}\leq e^{2t}\arctan(e^{-2t})\leq 2.$

Then

$|\chi_{D_{A}(e^{t}/2,e^{t})}-A_{t}(h_{A})(z,w)\pi e^{2t}|\leq 3.$

As

$e^{i}_{t}(\omega)=(\chi_{E^{i}_{t}}\cdot(\chi_{D_{A}(e^{t}_{n}/2,e^{t})}-\pi
e^{2t}(A_{t}h_{A})))^{SV}(\omega)$

and hence

$|e^{i}_{t_{n}}(\omega)|=e^{2t_{n}}O\left(\hat{\epsilon}^{1/N-2\delta}+(\frac{1}{\hat{\epsilon}})^{1+\delta}\left(\frac{\epsilon^{\prime}}{\hat{\epsilon}^{D+\alpha_{1}}}+\frac{e^{-\lambda
t_{n}}}{\epsilon^{\prime 2}\hat{\epsilon}^{2\alpha_{1}+2}}\right)\right).$

So taking $\epsilon^{\prime}=Ke^{-\nu t}$ with $\nu=\min(\lambda/3,2)$ and as
$\hat{\epsilon}^{2\alpha_{1}+2}$ and
$\hat{\epsilon}^{D+\alpha_{1}}\geq\hat{\epsilon}^{D+2\alpha_{1}+2}$, we have
for some $\lambda^{\prime}$

$|e^{i}_{t_{n}}(\omega)|=e^{2t_{n}}O\left(\hat{\epsilon}^{1/N-2\delta}+\left(\frac{e^{-\lambda^{\prime}t_{n}}}{\hat{\epsilon}^{D+2\alpha_{1}+3+\delta}}\right)\right)$

Finally taking $\hat{\epsilon}=e^{-f_{3}t}$ with
$f_{3}=\frac{\lambda^{\prime}}{D+2\alpha_{1}+3+\delta+1/N-2\delta}$ we have
shown that for each sequence $(t_{n})$ satisfying equation 4, for almost every
$\omega\in\mathcal{H}$ there is a $n_{0}(\omega)$ such that for $n\geq n_{0}$,
if $t_{n}>T_{0}$

$|e^{i}_{t_{n}}(\omega)|=e^{2t_{n}}O(e^{-f_{4}t})$

with $f_{4}>0$. For technical reason, which will appear after, we choose
$f_{4}<\min(f,2)$.

###  4.4 End of the estimation

Putting everything together we have for every sequence $(t_{n})$ respecting
condition of equation 4 for almost every $\omega\in\mathcal{H}$, and for every
$n\geq n_{0}(\omega)$,

$\displaystyle\left|N_{A}^{*}(\omega,e^{t_{n}})-\pi
e^{2t_{n}}\int_{\mathcal{H}}h_{A}d\mu\right|$
$\displaystyle\leq\left|N_{A}^{*}(\omega,e^{t_{n}})-\pi
e^{2t_{n}}(A_{t_{n}}\hat{h}_{A})(\omega)\right|+\left|\pi
e^{2t_{n}}\int_{\mathcal{H}}h_{A}d\mu-\pi
e^{2t_{n}}(A_{t}\hat{h}_{A})(\omega)\right|$
$\displaystyle\leq\left|N_{A}^{*}(\omega,e^{t_{n}})-\pi
e^{2t_{n}}(A_{t_{n}}\hat{h}_{A})(\omega)\right|+|m_{t_{n}}(\omega)|+\sum_{i=1}^{4}|e^{i}_{t_{n}}(\omega)|$
$\displaystyle=e^{2t_{n}}O(e^{-f_{4}t_{n}})$ (6)

We consider the family of sequence $(\frac{t_{n}}{2^{j}})$. For each sequence
there is a set of full measure $\mathcal{H}_{i}$ on which the estimate 4.4 is
true. Considering the set
$\mathcal{H}_{\infty}=\cap_{j\in\mathbb{N}}\mathcal{H}_{i}$, which is also of
full measure, we have the estimate 4.4 for all times $\frac{t_{n}}{2^{j}}$.

Fixing a natural number $n$, let’s call
$K=\lfloor\frac{t_{n}-ln(T_{0})}{ln(2)}\rfloor$ such that
$\frac{e^{t_{n}}}{2^{K}}>T_{0}$ we have

$\displaystyle\frac{N_{A}(\omega,e^{t_{n}})}{e^{2t_{n}}}$
$\displaystyle=\sum_{j=0}^{K}\frac{N_{A}^{*}(\omega,e^{t_{n}}/2^{j})}{e^{2t_{n}}}+\frac{N_{A}^{*}(\omega,e^{t_{n}}/2^{K})}{e^{2t_{n}}}$
$\displaystyle=\sum_{j=0}^{K}2^{-2j}\frac{N_{A}^{*}(\omega,e^{t_{n}}/2^{2j})}{e^{2t_{n}}/2^{2j}}+2^{-2K}\frac{N_{A}^{*}(\omega,e^{t_{n}}/2^{2K})}{e^{2t_{n}}/2^{2K}}$

As we have

$\frac{N_{A}^{*}(\omega,e^{t_{n}}/2^{2j})}{e^{2t_{n}}/2^{j}}=\pi\int_{\mathcal{H}}h_{A}d\mu+O\left(\frac{2^{f_{4}j}}{e^{f_{4}t_{n}}}\right)$

so

$\displaystyle\frac{N_{A}(\omega,e^{t_{n}})}{e^{2t_{n}}}$
$\displaystyle=\sum_{j=0}^{K}2^{-2j}\left(\pi\int_{\mathcal{H}}h_{A}d\mu+O\left(\frac{2^{f_{4}j}}{e^{f_{4}t_{n}}}\right)\right)+2^{-2K}\frac{N_{A}^{*}(\omega,e^{t_{n}}/2^{2K})}{e^{2t_{n}}/2^{K}}$
$\displaystyle=4(\frac{1-(1/4)^{K}}{3})\pi\int_{\mathcal{H}}h_{A}d\mu+O\left(\frac{1}{e^{f_{4}t_{n}}}\right)\frac{1-2^{(f_{4}-2)K}}{1-2^{f_{4}-2}}+2^{-2K}\frac{N_{A}^{*}(\omega,e^{t_{n}}/2^{2K})}{e^{2t_{n}}/2^{K}}$

By proposition 3.2 and equation 1.1 of [2]
$\frac{N_{A}^{*}(\omega,e^{t_{n}}/2^{2K})}{e^{2t_{n}}/2^{K}}$ is bounded by a
constant $M$. Moreover as $f_{4}<2$ we have also
$\frac{1-2^{(f_{4}-2)K}}{1-2^{f_{4}-2}}\leq M^{\prime}$

$\displaystyle|\frac{N_{A}(\omega,e^{t_{n}})}{e^{2t_{n}}}-c_{\mu}(A)|\leq
c_{\mu}(A)(1/4)^{K}+\frac{f(t_{n})}{e^{f_{4}t_{n}}}M^{\prime}+2^{-2K}M$

as $K\geq\frac{t_{n}-ln(T_{0})}{ln(2)}-1$ we have

$\displaystyle|\frac{N_{A}(\omega,e^{t_{n}})}{e^{2t_{n}}}-c_{\mu}(A)|$
$\displaystyle\leq
c_{\mu}(A)(1/4)^{\frac{t_{n}-ln(T_{0})}{ln(2)}-1}+\frac{f(t_{n})}{e^{f_{4}t_{n}}}M^{\prime}+2^{-2(\frac{t_{n}-ln(T_{0})}{ln(2)}-1)}M$
$\displaystyle=O(e^{-f_{5}t_{n}})$

with $f_{5}=\min(2\ln(2),f_{4})$.

####  4.4.1 Estimate for all times

Let’s take $t_{n}=\theta\log(n)$, which satisfy equation 4 for $\theta$ big
enough, by monotony we have for a general time $t$ such that $n\leq t\leq n+1$
and $n\geq T_{0}$,

$\displaystyle
e^{2\theta\log(n)}\left(c_{\mu}(A)-O(e^{-f_{5}\theta\log(n)})\right)\leq
N_{A}(\omega,e^{\theta\log(n)})\leq N_{A}(\omega,e^{\theta\log(t)})$
$\displaystyle\leq N_{A}(\omega,e^{\theta\log(n+1)})\leq
e^{2\theta\log(n+1)}\left(c_{\mu}(A)+O(e^{-f_{5}\theta\log(n+1)})\right)$

So that

$n^{2\theta}\left(c_{\mu}(A)-O(\frac{1}{n^{f_{5}\theta}})\right)\leq
N_{A}(\omega,t^{\theta})\leq(n+1)^{2\theta}\left(c_{\mu}(A)+O(\frac{1}{(n+1)^{f_{5}\theta}})\right)$

Then

$\left(\frac{n}{t}\right)^{2\theta}\left(c_{\mu}(A)-O\left(\left(\frac{t}{n}\right)^{f_{5}\theta}\frac{1}{t^{f_{5}\theta}}\right)\right)\leq\frac{N_{A}(\omega,t^{\theta})}{t^{2\theta}}\leq\left(\frac{n+1}{t}\right)^{2\theta}\left(c_{\mu}(A)+O\left(\left(\frac{t}{n+1}\right)^{f_{5}\theta}\frac{1}{t^{f_{5}\theta}}\right)\right)$

But then as $1-\frac{1}{t}\leq\frac{n}{t}\leq 1$ and $1\leq\frac{n+1}{t}\leq
1+\frac{1}{t}$ we have

$\left|\frac{N_{A}(\omega,t^{\theta})}{t^{2\theta}}-c_{\mu}(A)\right|=O\left(\frac{1}{t^{f_{5}\theta}}+\frac{1}{t^{2\theta}}\right)$

as $f_{5}\leq 2\ln(2)\leq 2$, taking $s=t^{\theta}$, we can conclude that

$\left|\frac{N_{A}(\omega,s)}{s^{2}}-c_{\mu}(A)\right|=O\left(\frac{1}{s^{f_{5}}}\right).$

This concludes the proof of the existence of the power saving error term in
Theorem 1.2.

##  5 Property of the constants $c_{\mu}(A)$

Fixing a $A>0$ and a ergodic $SL(2,\mathbb{R})$-invariant measure $\mu$, the
constant $c_{\mu}(A)$ remains mysterious. By the previous computation, it is
equal to

$c_{\mu}(A):=\frac{4}{3}\pi\int_{\mathcal{H}}\hat{h}_{A}d\mu=\frac{4}{3}\pi\int_{\mathbb{C}^{2}}h_{A}dm$

where $m$ is called _Siegel-Veech measure_ and is invariant with respect to
the $SL(2,\mathbb{R})$ action on $\mathbb{C}^{2}$. We can decompose further as

$\int_{\mathbb{C}^{2}}h_{A}dm=\int_{\mathbb{R}\setminus\\{0\\}}\left(\int_{SL(2,\mathbb{R})}h(tz,w)d\lambda(z,w)\right)d\nu(t)+\int_{\mathbb{P}^{1}(\mathbb{R})}\left(\int_{\mathbb{C}}h(z,sz)dz\right)d\rho(s)$

where $\lambda$ is the Haar measure on $SL(2,\mathbb{R})$ and $\nu=\nu(m)$ and
$\rho=\rho(m)$ are decomposition of the measure $m$, on which little is known.

We can still get a result of convergence for a $*$-weak limit of a sequence of
ergodic $SL(2,\mathbb{R})$-invariant probability measures.

######  Theorem 5.1.

Fixing an $A$, let $(\mu_{n})$ be a sequence of $SL(2,\mathbb{R})$-invariant
probability ergodic measures on $\mathcal{H}$, such that $\mu_{n}\to\nu$ in
the weak-$*$ topology, where $\nu$ is another ergodic
$SL(2,\mathbb{R})$-invariant probability measure. Then the constants satisfy

$c_{\mu_{n}}(A)\to c_{\nu}(A).$

The proof is similar to the proof for Siegel-Veech constants made by Dozier
[4]. We briefly recall it here.

###### Proof.

Let’s call $l(X)$ the length of the shortest saddle connection on the flat
surface $X$ and let $C_{K}=\\{X\in\mathcal{H},\frac{1}{l(X)}\leq K\\}$. We
take $\chi_{K}$ be a continuous function with image in $[0,1]$ whose value is
$1$ on $C_{K}$ and $0$ on $\mathcal{H}-C_{K+1}$.

Then

$c_{\mu_{n}}(A):=\frac{4}{3}\pi\int_{\mathcal{H}}\hat{h}_{A}d\mu_{n}=\frac{4}{3}\pi\left(\int_{\mathcal{H}}\hat{h}_{A}\chi_{K}d\mu_{n}+\int_{\mathcal{H}}\hat{h}_{A}(1-\chi_{K})d\mu_{n}\right)$

We want to use $*$-weak convergence of the measure for the integral on the
thick part of the stratum and to control the integral in the thin
independently of the measure $\mu_{n}$.

As $h_{A}$ is bounded and compactly supported, $\hat{h}_{A}$ is bounded by
$\hat{f}^{2}$ where $f=\chi_{B}\|h\|_{\infty}$ with $B$ a ball big enough to
contain the union of the projections of the support of $h$ via the maps
$\mathbb{R}^{4}\to\mathbb{R}^{2}$

$(x,y)\mapsto x\text{ and }(x,y)\mapsto y$

Eskin and Masur show that [6, Theorem 5.1] $\hat{f}<\frac{C}{l^{1+\delta}}$
for some $C$ and $0<\delta<1/2$. Choosing a $\delta^{\prime}$ in order that
$\delta+\delta^{\prime}<1/2$ we obtain

$\int_{\mathcal{H}}\hat{h}_{A}(1-\chi_{K})d\mu_{n}\leq\int_{\mathcal{H}-C_{K}}\frac{C}{l^{1+\delta}}d\mu_{n}\leq\frac{C}{K^{\delta^{\prime}}}\int_{\mathcal{H}-C_{K}}\frac{1}{l^{1+\delta+\delta^{\prime}}}d\mu_{n}$

The last integral is finite by a lemma of the two previously cited authors [6,
Lemma 5.5], and we can bound it from above by a constant independent of
$\mu_{n}$. First we recall a bound on circle average made by Dozier.

######  Lemma 5.2 (Proposition 1.1 of [4]).

For any stratum $\mathcal{H}$ and $0<\delta<1/2$, there exists a function
$\alpha:\mathcal{H}\to\mathbb{R}_{+}$ and constants $c_{0},b$ such that for
any $X\in\mathcal{H}$,

$\int_{0}^{2\pi}\frac{1}{l(g_{t}r_{\theta}X)^{1+\delta}}d\theta\leq
c_{0}e^{-(1-2\delta)t}\alpha(X)+b$

Then, choosing any smoothing function $\phi$ non-negative, smooth and
compactly supported with integral over $\mathbb{R}$ equal to $1$, we get by
Nevo ergodic Theorem 1.5, for a generic $X\in\mathcal{H}$

$\displaystyle\int_{\mathcal{H}-C_{K}}\frac{1}{l^{1+\delta+\delta^{\prime}}}d\mu_{n}$
$\displaystyle\leq\int_{\mathcal{H}}\frac{1}{l^{1+\delta+\delta^{\prime}}}d\mu_{n}$
$\displaystyle=\underset{t\to\infty}{\lim}\int_{-\infty}^{\infty}\phi(t-s)\left(\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{l(g_{s}r_{\theta}X)^{1+\delta}}d\theta\right)ds$
$\displaystyle\leq b$

So picking an $\epsilon$ we can choose $K$ big enough such that for all $n$

$\int_{\mathcal{H}}\hat{h}_{A}(1-\chi_{K})d\mu_{n}\leq\frac{C}{K^{\delta^{\prime}}}\int_{\mathcal{H}-C_{K}}\frac{1}{l^{1+\delta+\delta^{\prime}}}d\mu_{n}\leq\frac{Cb}{K^{\delta^{\prime}}}\leq\epsilon/2.$

Then by $*$-weak convergence of the measure there exist a $N$ such that for
$n\geq N$

$\int_{\mathcal{H}}\hat{h}_{A}\chi_{K}d\mu_{n}\leq\epsilon/2.$

Putting the two together gives the desired result. ∎

This theorem has a direct consequence on the possible values of the constants
$c_{\mu}(A)$ when $\mu$ varies on the space of ergodic
$SL(2,\mathbb{R})$-invariant probability measure, namely they all fall in a
closed interval of $\mathbb{R}_{>0}$. Using the following proposition of
Eskin, Mirzakhani, and Mohammadi

######  Theorem 5.3 (Theorem 2.3 of [8]).

The space of ergodic $SL(2,\mathbb{R})$-invariant probability measure is
compact in the $*$-weak topology.

We get the following corollary.

######  Corollary.

For every $A$, there are two constants $m(A),M(A)$ such that for every ergodic
$SL(2,\mathbb{R})$-invariant measure $\mu$

$m(A)\leq c_{\mu}(A)\leq M(A)$

###### Proof.

By contradiction taking a sequence of measure $\mu_{n}$ such that
$c_{\mu_{n}}(A)\to\infty$ we can extract a subsequence $*$-weakly converging
to a measure $\mu$. As the constant for this subsequence goes to $\infty$ and
to the constant of the limit measure, we have a contradiction.

The lower bound is obtained by a similar proof. ∎

###  5.1 End of proof of Theorem 1.2

Given an ergodic $SL(2,\mathbb{R})$-invariant measure $\mu$ on $\mathcal{H}$,
we showed in Section 3 the quadratic growth $N_{A}(\omega,R)$ for $\mu$-almost
every $\omega$ in $\mathcal{H}$. Then in Section 4 we proved that this
quadratic growth admits a power saving error term. Finally in this Section, we
demonstrated the continuity of the constants $c_{\mu}(A)$ with respect to
$\mu$. By putting these results together, we get a proof of the main result,
Theorem 1.2.

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Coend $(\mathbb{F},i)$ of $\mathcal{B}$ can be thought as universal right
comodule. Indeed, $\mathbb{F}$ coacts on each object $X$ of $\mathcal{B}$ via
the coaction $\phi_{X}\colon X\to X\otimes\mathbb{F}$, defined by setting

$\phi_{X}=(\mathrm{id}_{X^{*}}\otimes
i_{X})(\text{coev}_{X}\otimes\mathrm{id}_{X}),\quad\text{depicted
by}\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unimult1.eps}}}\;.$

This coaction is universal, that is, for any $D\in\mathrm{Ob}(\mathcal{B})$
and any natural transformation $\\{\gamma_{X}\colon X\to X\otimes
D\\}_{X\in\mathrm{Ob}(\mathcal{B})}$, there exists a unique morphism
$f\colon\mathbb{F}\to D$ such that for any $X\in\mathrm{Ob}(\mathcal{B})$,

$\gamma_{X}=(\mathrm{id}_{X}\otimes f)\phi_{X}.$

The structural morphisms of coend $\mathbb{F}$ are characterized in terms of
the universal coaction as follows: for all $X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\displaystyle\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unimult.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unimult1.eps}}}\;,\qquad
u_{\mathbb{F}}=\phi_{\mathbb{1}}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unimult1.eps}}}\;,$
$\displaystyle\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unidelta.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unidelta1.eps}}}\;,\qquad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unieps.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unieps1.eps}}},\;\qquad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unianti.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unianti1.eps}}}\;.$

Similarly, end $(\mathbb{A},j)$ of $\mathcal{B}$, where
$j=\\{j_{X}\colon\mathbb{A}\to X\otimes
X^{*}\\}_{X\in\mathrm{Ob}(\mathcal{B})}$ is the universal dinatural
transformation, can be thought as a universal left module, that is, for each
$X\in\mathrm{Ob}(\mathcal{B})$ there is a left action
$\alpha_{X}\colon\mathbb{A}\otimes X\to X$, defined by setting

$\alpha_{X}=(\mathrm{id}_{X}\otimes\text{ev}_{X})(j_{X}\otimes\mathrm{id}_{X}),\quad\text{depicted
by}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{mirunimult1.eps}}}.\;$

As above, this action is universal, that is, for any
$D\in\mathrm{Ob}(\mathcal{B})$ and any natural transformation
$\beta=\\{\beta_{X}\colon D\otimes X\to X\\}_{X\in\mathrm{Ob}(\mathcal{B})}$,
there is a unique morphism $f\colon D\to\mathbb{A}$ such that for any
$X\in\mathrm{Ob}(\mathcal{B})$,

$\beta_{X}=\alpha_{X}(f\otimes\mathrm{id}_{X}).$

The structural morphisms of end $\mathbb{A}$ (as in [BDR23, Section 7.1.]) are
characterized in terms of the universal action as follows: for all
$X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\displaystyle\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{mirunimult.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{mirunimult1.eps}}}\;,\qquad\varepsilon_{\mathbb{A}}=\alpha_{\mathbb{1}}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{mirunimult1.eps}}}\;,$
$\displaystyle\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{mirunidelta.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{mirunidelta1.eps}}}\;,\qquad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{mirunieps.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unieps1.eps}}},\;\qquad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{mirunianti.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{mirunianti1.eps}}}\;.$

Note that if a coend of a braided pivotal category exists, then an end of it
also exists and it is realized as dual of its coend. See [FGR19, Lemma 4.12.]
for details.

###  9.2. Drinfeld map

By rewriting [FGR19, equation (4.17)] via (co)actions of (co)ends, there is a
unique map $D\colon\mathbb{F}\to\mathbb{A}$, called Drinfeld map, such that
for all $X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld1.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld2.eps}}}.\;$

The Drinfeld map is a morphism of Hopf algebras $\mathbb{F}$ and $\mathbb{A}$.
Indeed, for all $X,Y,Z\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld3.eps}}}\;\overset{(i)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld4.eps}}}\overset{(ii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld5.eps}}}\overset{(iii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld6.eps}}}\overset{(iv)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld7.eps}}}.\;$

Here $(i)$ follows by definition of $m_{\mathbb{F}}$ and
$D_{\mathbb{F},\mathbb{A}}$, $(ii)$ by isotopy, $(iii)$ by definition of
$D_{\mathbb{F},\mathbb{A}}$ and isotopy, and $(iv)$ follows by definition of
$m_{\mathbb{A}}$.

Next, for all $X\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld9.eps}}}\overset{(i)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld1.eps}}}\;\overset{(ii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{drinfeld2.eps}}}\overset{(iii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unieps1copy.eps}}}\;\overset{(iv)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{uniunitcopy.eps}}}\;.$

Here, $(i)$ (respectively, $(iv)$) follows by definition of $u_{\mathbb{F}}$
(respectively, $u_{\mathbb{A}}$), $(ii)$ by definition of
$D_{\mathbb{F},\mathbb{A}}$, $(iii)$ by the fact that
$\tau_{\mathbb{1},X}=\mathrm{id}_{X}$ for any $X\in\mathrm{Ob}(\mathcal{B})$.
In conclusion, $D_{\mathbb{F},\mathbb{A}}$ is an algebra morphism. One
verifies similarly that it is also a coalgebra morphism.

######  Remark 9.1.

A unimodular ribbon category $\mathcal{B}$ is factorizable (see [FGR19]) if
the morphism

$D_{\mathbb{F}}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{Dfeld0.eps}}}\;\colon\mathbb{F}\to\mathbb{F}^{*},$

where $\omega$ is the Hopf pairing on $\mathbb{F}$ (see Section 5.19), is an
isomorphism. The latter is equivalent to the non-degeneracy of $\omega$ in the
sense of Section 5.18. Indeed, if $D_{\mathbb{F}}$ is an isomorphism, then
$\Omega=D_{\mathbb{F}}^{-1}\text{coev}_{\mathbb{F}}$ is an inverse of
$\omega$. Conversely, if $\omega$ is non-degenerate with inverse $\Omega$,
then $D_{\mathbb{F}}$ is an isomorphism with inverse
$D_{\mathbb{F}}^{-1}=(\text{ev}_{\mathbb{F}}\otimes\mathrm{id}_{\mathbb{F}})(\mathrm{id}_{\mathbb{F}^{*}}\otimes\Omega)$.

According to [BDR23, Appendix A.], the factorizability or modularity of in the
sense of [BDR23] is equivalent to the notion of factorizability in the above
sense. The latter is also equivalent to invertibility of Drinfeld map (see
[FGR19, Proposition 4.11.]).

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