2.1 Outline

In this section, we discuss horospheres on connected components of strata of quadratic differentials from several points of view. After defining the horospherically foliated sets $\mathcal {Q}(\gamma )$ , we parameterize them using a special class of cylinder diagrams. We then discuss how to parameterize them in terms of singular measured foliations and, more concretely, how to do so using train track coordinates. This last perspective will be particularly useful in the proof of Theorem 1.1.

2.2 Horospheres

For the rest of this section, fix an integer vector $\sigma := (\sigma _1,\ldots ,\sigma _n)$ with $\sigma _i \geq -1$ and a boolean value $\varepsilon \in \{0,1\}$ . Denote by $\mathcal {Q}(\sigma ,\varepsilon )$ the stratum of unmarked quadratic differentials with marked singularities of order $\sigma $ and $\varepsilon = 1$ if and only if every quadratic differential in the stratum is the square of an Abelian differential. A singularity of order $-1$ is a pole, a singularity of order $0$ is a marked point, and a singularity of order $\geq 1$ is a zero of the corresponding order. Let $g \geq 0$ be the non-negative integer satisfying the equation

$$ \begin{align*} 4g-4 = \sum_{i =1}^n \sigma_i. \end{align*} $$

The complex dimension of the stratum $\mathcal {Q}(\sigma ,\varepsilon )$ is given by

$$ \begin{align*} h:= 2g-2+n+\epsilon. \end{align*} $$

For the rest of this section, we fix a connected component $\mathcal {Q}$ of this stratum.

Let S be a connected, oriented surface of genus g with n punctures. Denote by $\mathcal {Q}\mathcal {T}$ the lift of the connected component $\mathcal {Q}$ of quadratic differentials marked by the surface S. Denote by $\mathrm {Mod}:= \mathrm {Mod}(S)$ the mapping class group of S. The action of $\mathrm {Mod}$ on $\mathcal {Q}\mathcal {T}$ is properly discontinuous and $\mathcal {Q}$ is the corresponding orbifold quotient.

Denote by $\mathcal {MF} := \mathcal {MF}(S)$ the space of singular measured foliations on S. The markings on $Q\mathcal {T}$ allow us to define maps

$$ \begin{align*} \Re, \Im \colon Q\mathcal{T} \to \mathcal{MF} \end{align*} $$

that record the vertical and horizontal foliations, respectively, of any marked quadratic differential. Fix an integral simple closed multi-curve $\gamma := a_1\gamma _1+\cdots +a_k\gamma _k$ on S. Any such multi-curve determines a horospherically foliated set

$$ \begin{align*} \mathcal{Q}\mathcal{T}(\gamma) := \{q \in \mathcal{QT} \ | \ \Im(q) = \gamma\}. \end{align*} $$

The maps $\Re $ and $\Im $ above factor through the mapping class group action to give

$$ \begin{align*} \boldsymbol{\Re}, \boldsymbol{\Im} \colon \mathcal{Q} \to \mathcal{MF}/\mathrm{Mod}. \end{align*} $$

Equivalence classes of integral simple closed multi-curves are called topological types. Fixing a topological type gives rise to a horospherically foliated set

$$ \begin{align*} \mathcal{Q}(\gamma) := \{q \in \mathcal{Q} \ | \ \boldsymbol{\Im}(q) = \mathrm{Mod}\cdot\gamma\}. \end{align*} $$

Denote by $\mathrm {Stab}(\gamma ) \subseteq \mathrm {Mod}$ the set of mapping classes that fix $\gamma $ . It will be convenient for our purposes to consider the intermediate quotient $\mathcal {QT}(\gamma )/ \mathrm {Stab}(\gamma )$ . The following proposition will be particularly useful in this context.

2.3 Cylinder diagrams

Quadratic differentials in $\mathcal {Q}(\gamma )$ can be represented via cylinder diagrams. A cylinder diagram is a collection of disjoint parallelograms on the complex plane with certain edge identifications. Each parallelogram has one pair of sides, called bases, which are parallel to the real axis. These bases are broken into edges, and pairs of edges are identified by translation and/or 1800 rotation. The other pair of sides, referred to as special edges, are always identified by translation. See Figure 2 for an example.

Figure 2 Cylinder diagram of a quadratic differential in $\mathcal {Q}(4,0)$ . The horizontal foliation is identified with a simple closed curve on a genus $0$ surface with four punctures that separates the punctures into two sets of two. This is a moderately slanted cylinder diagram.

The horizontal foliation of a cylinder diagram is identified with a weighted simple closed multi-curve on the corresponding surface. The components of the multi-curve are given by the core curves of the cylinders and each such curve is weighted by the height of the corresponding cylinder. In the case of cylinder diagrams coming from integral simple closed multi-curves, the heights of the cylinders are always integers.

Each cylinder diagram represents a quadratic differential in a given connected component of a stratum. As we vary the real parts of the edges of a cylinder diagram while retaining the parallelism conditions, the corresponding quadratic differential remains in the same connected component. The identifications in the moduli space $\mathcal {Q}(\gamma )$ correspond to cut and paste operations.

To get a more complete description of the correspondence in Proposition 2.2, we restrict our attention to a particular class of cylinder diagrams. Consider a cylinder diagram with pairs of bases of lengths $b_1,\ldots ,b_k> 0$ and special edges of holonomy with real parts $s_1,\ldots ,s_k \in \mathbb {R}$ . We say this cylinder diagram is moderately slanted if $0 < s_i \leq b_i$ for every $i \in \{1,\ldots ,k\}$ . See Figure 2 for an example. The following proposition follows immediately by applying Dehn twists in an appropriate way.

In particular, we deduce the following corollary.

2.4 Measured foliations

Quadratic differentials in $\mathcal {Q}(\gamma )$ can also be parameterized in terms of their vertical foliations. More precisely, denote by $i(\cdot ,\cdot )$ the geometric intersection number pairing on $\mathcal {MF} \times \mathcal {MF}$ . Consider the set $\Delta \subseteq \mathcal {MF} \times \mathcal {MF}$ of pairs of non-filling singular measured foliations, that is,

$$ \begin{align*} \Delta := \{(\unicode{x3bb},\mu) \in \mathcal{MF} \times \mathcal{MF} \ | \ \text{there exists } \eta \in \mathcal{MF} \colon i(\unicode{x3bb},\eta) + i(\mu,\eta) = 0 \}. \end{align*} $$

By the work of Hubbard and Masur [Reference Hubbard and MasurHM79], the map

$$ \begin{align*} (\Re,\Im) \colon \mathcal{Q}\mathcal{T} \to \mathcal{MF} \times \mathcal{MF} \setminus \Delta \end{align*} $$

is a $\mathrm {Mod}$ -equivariant homeomorphism onto its image. Denote

$$ \begin{align*} \mathcal{MF}(\gamma) := \{\unicode{x3bb} \in \mathcal{MF} \ | \ (\gamma,\unicode{x3bb}) \notin \Delta\}. \end{align*} $$

Of particular importance for us will be the quotient $\mathcal {MF}(\gamma )/\mathrm {Stab}(\gamma )$ . Directly from Proposition 2.5, we deduce the following corollary.

2.5 Train tracks

The correspondence in Proposition 2.5 and Corollary 2.6 will allow us to study the quotient $\mathcal {MF}(\gamma )/\mathrm {Stab}(\gamma )$ using moderately slanted cylinder diagrams. We can make this idea more precise using train tracks. Given a moderately slanted cylinder diagram, consider a triangulation by saddle connections of the underlying surface S as in Figure 3.

Figure 3 Triangulation associated to the moderately slanted cylinder diagram in Figure 2.

On each of the triangles of this triangulation, consider a 1-complex as in Figure 4(a); the edges of this complex that do not intersect the sides of the triangle will be referred to as inner edges. Label the edges of the triangle by $a, b, c$ so that

$$ \begin{align*} |\Re(a)| = |\Re(b)| + |\Re(c)|. \end{align*} $$

The edge labeled a is unique because the cylinder diagram is moderately slanted. Delete the inner edge of the complex of the triangle opposite to a as in Figure 4(b). Joining these complexes along the edges of the triangulation, as in Figure 5, yields a train track $\tau $ on S that carries the singular measured foliation $\Re (q)$ ; the weights of the train track correspond to the absolute value of the real parts of the edges of the triangulation. Furthermore, the area of q is equal to $i(\Re (q),\gamma )$ .

Figure 4 The $1$ -complexes in a triangle.

Figure 5 Train track associated to the moderately slanted cylinder diagram in Figure 2.

We refer to the train tracks constructed above as moderately slanted cylinder train tracks. Varying the real parts of the edges of the cylinder diagram while retaining the parallelism and moderately slanted conditions preserves the train track. Moreover, the moderately slanted condition corresponds to an explicit linear cone in the weight space of the train track. We refer to this cone as the characteristic cone of the train track. Directly from this discussion, Proposition 2.1, and Corollaries 2.4 and 2.6, we deduce the following result.

Recall $\gamma := a_1\gamma _1 + \cdots + a_k \gamma _k$ . Consider a moderately slanted cylinder diagram with k cylinders $C_1,\ldots ,C_k$ representing a quadratic differential $q \in \mathcal {Q}(\gamma )$ . Denote by $\tau $ the corresponding moderately slanted cylinder train track. Let $u_1,\ldots ,u_k> 0$ be the weights on $\tau $ corresponding to the special edges of each cylinder. For each cylinder $C_i$ , let $v_1^{(i)},\ldots ,v_{m(i)}^{(i)}> 0$ be the weights on $\tau $ corresponding to the edges on the top base of the cylinder ordered from left to right. For each cylinder $C_i$ , let $w_1^{(i)},\ldots ,w_{l(i)}^{(i)}> 0$ be the weights on $\tau $ corresponding to the edges on the bottom base of the cylinder ordered from left to right. The weights considered completely determine the rest of the weights of the $\tau $ . Notice that

$$ \begin{align*} \mathrm{Area}(q) := \frac{1}{2}\bigg(\sum_{i =1}^k a_i \bigg(\sum_{j=1}^{m(i)} v_j^{(i)} + \sum_{j=1}^{l(i)} w_j^{(i)}\bigg)\bigg). \end{align*} $$

Directly from the discussion above, we deduce the following.

3.1 Outline of this section

In this section, we prove Theorem 1.1, the main result of this paper. The proof relies on the sophisticated theory developed in the work of Eskin, Mirzakhani, and Mohammadi [Reference Eskin, Mirzakhani and MohammadiEMM22] for counting mapping class group orbits of simple closed multi-curves in train track coordinates. We begin by reviewing this theory and discussing some variants. We then apply this theory together with the result discussed in §2 to prove Theorem 1.1.

3.2 Mapping class group orbits

In [Reference Eskin, Mirzakhani and MohammadiEMM22], Eskin, Mirzakhani, and Mohammadi proved an effective estimate for the number of simple closed curves of length $\leq L$ on a compact surface equipped with a Riemannian metric of negative curvature. To prove this result, a sophisticated theory for counting mapping class group orbits of integral simple closed multi-curves in train track coordinates was developed. We now summarize the main aspects of this theory as well as discuss some variants.

Let S be a connected, oriented surface of genus g with n punctures and $\tau $ be a train track on S. Denote by $U(\tau )$ the cone of non-negative weights on $\tau $ satisfying the switch conditions and by $\|\cdot \|$ the $L^1$ norm on $U(\tau )$ . Consider the set

$$ \begin{align*} P(\tau) := \{\unicode{x3bb} \in U(\tau) \ | \ \|\unicode{x3bb}\| = 1\}. \end{align*} $$

By a polyhedron $\mathcal {U} \subseteq P(\tau )$ , we mean a polyhedron of dimension $\mathrm {dim} \thinspace U(\tau ) -1$ , where the number of facets and the angles are bounded below by uniform constants depending only on S; facets are allowed to be open and/or closed. For the rest of this discussion, let $\gamma := a_1\gamma _1 + \cdots + a_k \gamma _k$ be an integral simple closed multi-curve on S. For every $L \geq 0$ , consider the counting function

$$ \begin{align*} s(\gamma,\mathcal{U},L) := \#\left\lbrace \begin{array}{@{}c | l@{}} \alpha \in \mathrm{Mod}(S) \cdot \gamma & \alpha \in \mathbb{R}^+ \cdot \mathcal{U} \\ & \|\alpha\| \leq L \end{array}\right\rbrace\!. \end{align*} $$

Let $\mathcal {Q}$ be the principal stratum of quadratic differentials on S. Denote $v(\gamma )> 0$ the Lebesgue measure of the set of quadratic differentials $q \in Q(\gamma )$ with $\mathrm {Area}(q) \leq 1$ ; this quantity is finite because of Propositions 2.7 and 2.8. Denote by $\mu _{\mathrm {Thu}}$ the Thurston measure on $\mathcal {MF}(S)$ . Recall that $\mathcal {MF}(S)$ can be endowed with a natural $\mathbb {R}^+$ action that scales transverse measures.

Theorem 3.1. [Reference Eskin, Mirzakhani and MohammadiEMM22, Theorem 7.1]

There exists $\kappa = \kappa (S)> 0$ such that for every maximal train track $\tau $ on S, every polyhedron $\mathcal {U} \subseteq U(\tau )$ , and every $L \geq 0$ ,

$$ \begin{align*} s(\gamma,\mathcal{U},L) = v(\gamma) \cdot \mu_{\mathrm{Thu}}((0,1]\cdot \mathcal{U}) \cdot L^{6g-6} + O_{\gamma,\tau}(L^{6g-6-\kappa}). \end{align*} $$

In the ensuing discussion, we use the notation introduced in §2. Let $\mathcal {Q}$ be a connected component of a stratum of quadratic differentials. Recall that we denote by $h> 0$ its complex dimension. Denote by $v(\gamma ,\mathcal {Q})> 0$ the Lebesgue measure of the set of quadratic differentials $q \in Q(\gamma )$ with $\mathrm {Area}(q) \leq 1$ ; this quantity is finite because of Propositions 2.7 and 2.8. Given a moderately slanted cylinder train track $\tau $ , denote by $\mu $ the Lebesgue measure on its weight space.

Theorem 3.2. There exists $\kappa = \kappa (\mathcal {Q})> 0$ such that for every moderately slanted cylinder train track $\tau $ on S carrying vertical foliations of quadratic differentials in $\mathcal {Q}(\gamma )$ , every polyhedron $\mathcal {U} \subseteq U(\tau )$ , and every $L \geq 0$ ,

$$ \begin{align*} s(\gamma,\mathcal{U},L) = v(\gamma,\mathcal{Q}) \cdot \mu((0,1]\cdot \mathcal{U}) \cdot L^h + O_{\gamma,\tau}(L^{h-\kappa}). \end{align*} $$

3.3 Square-tiled surfaces

For the rest of this section, fix $\mathcal {Q}$ a connected component of a stratum of quadratic differentials on a surface S, and integral simple closed multi-curves $\gamma _1$ and $\gamma _2$ on S. For every $L \geq 0$ , consider the counting function

$$ \begin{align*} sq(\gamma_1,\gamma_2,\mathcal{Q},L) := \#\left(\left\lbrace \begin{array}{@{}c|l@{}} \text{square-tiled surfaces }q & q \in \mathcal{Q},\\ & \boldsymbol{\Re}(q) = \mathrm{Mod} \cdot \gamma_1, \\ & \boldsymbol{\Im}(q) = \mathrm{Mod} \cdot \gamma_2, \\ & \mathrm{Area}(q) \leq L. \end{array} \right\rbrace \bigg/ \sim\right)\!, \end{align*} $$

where $\sim $ denotes the equivalence relation induced by cut-and-paste operations.

Our goal for the rest of this section is to prove an effective estimate for this counting function. To do so, we first recast it as a counting function of mapping class group orbits of integral simple closed multi-curves in train track coordinates. We then apply Theorem 3.2 to get the desired effective estimate.

3.4 Recasting

By the work of Hubbard and Masur [Reference Hubbard and MasurHM79], square-tiled surfaces are in one-to-one correspondence with filling pairs of integral simple closed multi-curves. Given a filling pair of integral simple closed multi-curves $\alpha $ and $\beta $ on S, denote by $q(\alpha ,\beta )$ the corresponding square-tiled surface. Notice that

$$ \begin{align*} \mathrm{Area}(q(\alpha,\beta)) = i(\alpha,\beta). \end{align*} $$

Using this correspondence, we can recast the counting function $sq(\gamma _1,\gamma _2,\mathcal {Q},L)$ in a more convenient way; compare with [Reference Arana-HerreraAH20, §3].

Proposition 3.3. In the context above, for every $L \geq 0$ ,

$$ \begin{align*} sq(\gamma_1,\gamma_2,\mathcal{Q},L) =\#\left(\left\lbrace \begin{array}{@{}c | l@{}} \alpha \in \mathrm{Mod} \cdot \gamma_1 & \alpha \in \mathcal{MF}(\gamma_2),\\ & q(\alpha,\gamma_2) \in \mathcal{Q}, \\ & i(\alpha,\gamma_2) \leq L, \end{array} \right\rbrace \bigg/ \mathrm{Stab}(\gamma_2)\right)\!. \end{align*} $$

Proof. Using the correspondence between square-tiled surfaces and filling pairs of integral simple closed multi-curves, we write

$$ \begin{align*} sq(\gamma_1,\gamma_2,\mathcal{Q},L) =\#\left(\left\lbrace \begin{array}{@{}c | l@{}} (\alpha,\beta) \in \mathrm{Mod} \cdot \gamma_1 \times \mathrm{Mod} \cdot \gamma_2 & (\alpha,\beta) \notin \Delta,\\ & q(\alpha,\beta) \in \mathcal{Q}, \\ & i(\alpha,\beta) \leq L, \end{array} \right\rbrace \bigg/ \mathrm{Mod}\right)\!, \end{align*} $$

where the quotient by $\mathrm {Mod}$ corresponds to the diagonal action on $\mathcal {MF} \times \mathcal {MF}$ . In turn, this expression can be rewritten as

$$ \begin{align*} sq(\gamma_1,\gamma_2,\mathcal{Q},L) =\#\left(\left\lbrace \begin{array}{@{}c | l@{}} \alpha \in \mathrm{Mod} \cdot \gamma_1 & \alpha \in \mathcal{MF}(\gamma_2),\\ & q(\alpha,\gamma_2) \in \mathcal{Q}, \\ & i(\alpha,\gamma_2) \leq L, \end{array} \right\rbrace \bigg/ \mathrm{Stab}(\gamma_2)\right)\!.\\[-46pt] \end{align*} $$

3.5 Geometric intersection numbers

Fix $\tau $ a moderately slanted cylinder train track carrying singular measured foliations in $\mathcal {MF}(\gamma _2)$ corresponding to quadratic differentials in $\mathcal {Q}(\gamma _2)$ . Proposition 2.8 guarantees the function $i(\cdot ,\gamma _2)$ is a linear function over the characteristic cone of $\tau $ . In particular, this function is Lipschitz. Given a non-empty polyhedron $\mathcal {U} \subseteq P(\tau )$ in the characteristic cone of $\tau $ , denote

$$ \begin{gather*} M(\gamma_2,\mathcal{U}) := \max_{\unicode{x3bb} \in\kern1pt \mathcal{U}} i(\unicode{x3bb},\gamma_2), \\m(\gamma_2,\mathcal{U}) := \min_{\unicode{x3bb} \in\kern1pt \mathcal{U}} i(\unicode{x3bb},\gamma_2). \end{gather*} $$

Proposition 2.8 ensures these quantities are positive and finite. Denote the $L^1$ diameter of the polyhedron $\mathcal {U}$ by $\mathrm {diam}(\mathcal {U})$ . As a direct consequence of the discussion above, we deduce the following.

Proposition 3.4. For every polyhedron $\mathcal {U} \subseteq P(\tau )$ in the characteristic cone of $\tau $ ,

$$ \begin{align*} |M(\gamma_2,\mathcal{U}) - m(\gamma_2,\mathcal{U})| \preceq_{\gamma_2,\tau} \mathrm{diam}(\mathcal{U}). \end{align*} $$

3.6 Comparison

For every non-empty polyhedron $\mathcal {U} \subseteq P(\tau )$ in the characteristic cone of $\tau $ and every $L \geq 0$ , consider the counting function

$$ \begin{gather*} sq(\gamma_1,\gamma_2,\mathcal{U},L) :=\#\left\lbrace \begin{array}{@{}c | l@{}} \alpha \in \mathrm{Mod} \cdot \gamma_1 & \alpha \in \mathbb{R}_+ \cdot \mathcal{U}\\ & i(\alpha,\gamma_2) \leq L \end{array} \right\rbrace\!. \end{gather*} $$

When $\mathcal {U} \subseteq P(\tau )$ is the intersection of the characteristic cone of $\tau $ with $P(\tau )$ , we denote this counting function simply by $s(\gamma _1,\gamma _2,\tau ,L)$ . By Proposition 3.3, a first step toward the proof of Theorem 1.1, the main result of this paper, would be to study the counting functions $sq(\gamma _1,\gamma _2,\mathcal {U},L)$ . Theorem 3.2 allows us to study the counting functions $s(\gamma _1,\mathcal {U},L)$ . The following bounds, which follow directly from the definitions, will thus play an important role.

Proposition 3.5. For every non-empty polyhedrom $\mathcal {U} \subseteq P(\tau )$ in the characteristic cone of $\tau $ and every $L \geq 0$ , the following bounds hold:

$$ \begin{gather*} sq(\gamma_1,\gamma_2,\mathcal{U},L) \leq s(\gamma_1,\mathcal{U},L/m(\gamma_2,\mathcal{U})), \\ s(\gamma_1,\mathcal{U},L/M(\gamma_2,\mathcal{U})) \leq sq(\gamma_1,\gamma_2,\mathcal{U},L). \end{gather*} $$

Applying Theorem 3.2, we deduce the following corollary.

Corollary 3.6. There exists a constant $\kappa = \kappa (\mathcal {Q})> 0$ such that for every non-empty polyhedrom $\mathcal {U} \subseteq P(\tau )$ in the characteristic cone of $\tau $ and every $L \geq 0$ ,

$$ \begin{gather*} sq(\gamma_1,\gamma_2,\mathcal{U},L) \leq v(\gamma_1,\mathcal{Q}) \cdot \mu((0,1]\cdot \mathcal{U}) \cdot (L/m(\gamma_2,\mathcal{U}))^h + O_{\gamma_1,\tau}(L^{h-\kappa}) , \\ v(\gamma_1,\mathcal{Q}) \cdot \mu((0,1]\cdot \mathcal{U}) \cdot (L/M(\gamma_2,\mathcal{U}))^h + O_{\gamma_1,\tau}(L^{h-\kappa}) \leq sq(\gamma_1,\gamma_2,\mathcal{U},L). \end{gather*} $$

3.7 Leading terms

Let $\mathcal {V} \subseteq P(\tau )$ be the intersection of the characteristic cone of $\tau $ with $P(\tau )$ . Given a finite partition $\mathcal {U} := \{\mathcal {U}_i\}_{i=1}^N$ of $\mathcal {V}$ , denote

$$ \begin{align*} \mathrm{diam}(\mathcal{U}) := \max_{i \in \{1,\ldots,N\}} \mathrm{diam}(\mathcal{U}_i). \end{align*} $$

Denote by $v(\gamma _2,\tau )> 0$ the positive constant

$$ \begin{align*} v(\gamma_2,\tau) := \mu (\{\unicode{x3bb} \in \mathbb{R}^+ \cdot \mathcal{V} \ | \ i(\unicode{x3bb},\gamma_2) \leq 1 \} ). \end{align*} $$

The following proposition will allow us to identify the leading term of the counting function $s(\gamma _1,\gamma _2,\tau ,L)$ in the estimates that will be carried out later.

Proposition 3.7. For every partition $\mathcal {U} := \{\mathcal {U}_i\}_{i=1}^N$ of $\mathcal {V}$ ,

$$ \begin{gather*} \sum_{i=1}^N \frac{\mu((0,1]\cdot \mathcal{U}_i)}{M(\gamma_2,\mathcal{U}_i)^h} \leq v(\gamma_2,\tau) \leq \sum_{i=1}^N \frac{\mu((0,1]\cdot \mathcal{U}_i)}{m(\gamma_2,\mathcal{U}_i)^h}, \\ \sum_{i=1}^N \frac{\mu((0,1]\cdot \mathcal{U}_i)}{m(\gamma_2,\mathcal{U}_i)^h} - \sum_{i=1}^N \frac{\mu((0,1]\cdot \mathcal{U}_i)}{M(\gamma_2,\mathcal{U}_i)^h} \preceq_{\gamma_2,\tau} \mathrm{diam}(\mathcal{U}). \end{gather*} $$

Proof. The first set of bounds follows directly from the definitions and the fact that the measure $\mu $ is h-homogeneous under positive scalings. For the second bound, notice that by applying Proposition 3.4,

$$ \begin{align*} \sum_{i=1}^N \frac{\mu((0,1]\cdot \mathcal{U})}{m(\gamma_2,\mathcal{U})^h} &- \sum_{i=1}^N \frac{\mu((0,1]\cdot \mathcal{U})}{M(\gamma_2,\mathcal{U})^h}\\ &\preceq_{\gamma_2,\tau} \mu((0,1] \cdot \mathcal{V}) \cdot \max_{i \in \{1,\ldots,N\}}|M(\gamma_2,\mathcal{U}_i) - m(\gamma_2,\mathcal{U}_i)|\\ &\preceq_{\gamma_2,\tau} \mathrm{diam}(\mathcal{U}).\\[-3pc] \end{align*} $$

3.8 Characteristic cones

We are now ready to prove an effective estimate for the counting function $sq(\gamma _1,\gamma _2,\tau ,L)$ . This will be the main tool used in the proof of Theorem 1.1, the main result of this paper.

Proposition 3.8. There exists a constant $\kappa = \kappa (\mathcal {Q})> 0$ such that for every $L \geq 0$ ,

$$ \begin{align*} sq(\gamma_1,\gamma_2,\tau,L) = v(\gamma_1,\mathcal{Q}) \cdot v(\gamma_2,\tau) \cdot L^h + O_{\gamma_1,\gamma_2,\tau}(L^{h-\kappa}). \end{align*} $$

Proof. Let $\delta \in (0,1)$ to be fixed later. Consider a partition $\mathcal {U} := \{\mathcal {U}_i\}_{i=1}^N$ of diameter $\mathrm {diam}(\mathcal {U}) \leq \delta $ of the characteristic cone of $\tau $ into $N \preceq _{\tau } \delta ^{-h}$ polyhedrons. By Corollary 3.6, for each of these polyhedrons, we have the estimates

$$ \begin{gather*} sq(\gamma_1,\gamma_2,\mathcal{U}_i,L) \leq v(\gamma_1,\mathcal{Q}) \cdot \mu((0,1]\cdot \mathcal{U}_i) \cdot (L/m(\gamma_2,\mathcal{U}_i))^h + O_{\gamma_1,\tau}(L^{h-\kappa}) , \\ v(\gamma_1,\mathcal{Q}) \cdot \mu((0,1]\cdot \mathcal{U}_i) \cdot (L/M(\gamma_2,\mathcal{U}_i))^h + O_{\gamma_1,\tau}(L^{h-\kappa}) \leq sq(\gamma_1,\gamma_2,\mathcal{U}_i,L). \end{gather*} $$

Adding up these estimates over $i \in \{1,\ldots ,N\}$ , we get

$$ \begin{gather*} sq(\gamma_1,\gamma_2,\tau,L) \leq v(\gamma_1,\mathcal{Q}) \cdot \sum_{i=1}^N \frac{\mu((0,1]\cdot \mathcal{U}_i)}{m(\gamma_2,\mathcal{U}_i)^h} \cdot L^h + O_{\gamma_1,\tau}(N \cdot L^{h-\kappa}) , \\ v(\gamma_1,\mathcal{Q}) \cdot \sum_{i=1}^N \frac{\mu((0,1]\cdot \mathcal{U}_i)}{M(\gamma_2,\mathcal{U}_i)^h} \cdot L^h + O_{\gamma_1,\tau}(N \cdot L^{h-\kappa}) \leq sq(\gamma_1,\gamma_2,\mathcal{U}_i,L). \end{gather*} $$

By Proposition 3.7, it follows that

$$ \begin{align*} sq(\gamma_1,\gamma_2,\tau,L) = v(\gamma_1,\mathcal{Q}) \cdot v(\gamma_2,\tau) \cdot L^h+ O_{\gamma_1,\gamma_2,\tau}(\mathrm{diam}(\mathcal{U}) \cdot L^h + N \cdot L^{h-\kappa} ). \end{align*} $$

Notice that

$$ \begin{align*} \mathrm{diam}(\mathcal{U}) \cdot L^h + N \cdot L^{h-\kappa} \preceq_{\tau} \delta \cdot L^h + \delta^{-h} \cdot L^{h-\kappa}. \end{align*} $$

Let $\delta := L^{-\eta }$ with $\eta> 0$ . Choose $\eta> 0$ so that

$$ \begin{align*} \kappa'=\kappa'(\mathcal{Q}) := \min\{\eta, \kappa - h\eta\}> 0. \end{align*} $$

It follows that

$$ \begin{align*} sq(\gamma_1,\gamma_2,\tau,L) = v(\gamma_1,\mathcal{Q}) \cdot v(\gamma_2,\tau) \cdot L^h+ O_{\gamma_1,\gamma_2,\tau}(L^{h-\kappa'}). \\[-34.5pt] \end{align*} $$

3.9 Proof of the main result

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1

By Proposition 2.7, there exists a finite collection $\{\tau _i\}_{i=1}^N$ of moderately slanted cylinder train tracks which, in their characteristic cone, carry all singular measured foliation in the image of $\Re \colon \mathcal {QT}(\gamma _2) \to \mathcal {MF}(\gamma _2)$ up to the action of $\mathrm {Stab}(\gamma _2)$ without overlaps. By Proposition 3.3, we have

$$ \begin{align*} sq(\gamma_1,\gamma_2,\mathcal{Q},L) = \sum_{i=1}^N sq(\gamma_1,\gamma_2,\tau_i,L). \end{align*} $$

By Proposition 3.8, for every $i \in \{1,\ldots ,N\}$ ,

$$ \begin{align*} sq(\gamma_1,\gamma_2,\tau_i,L) = v(\gamma_1,\mathcal{Q}) \cdot v(\gamma_2,\tau_i) \cdot L^h+ O_{\gamma_1,\gamma_2,\tau_i}(L^{h-\kappa}). \end{align*} $$

Adding up these estimates, we conclude

$$ \begin{align*} sq(\gamma_1,\gamma_2,\mathcal{Q},L) = v(\gamma_1,\mathcal{Q}) \cdot v(\gamma_2,\mathcal{Q}) \cdot L^h+ O_{\gamma_1,\gamma_2,\mathcal{Q}}(L^{h-\kappa}).\\[-34.5pt] \end{align*} $$

3.10 Further remarks

As explained in §1, the ideas introduced in the proof of Theorem 1.1 can also be used to give effective estimates of other related counting functions of square-tiled surfaces. More explicitly, for every $L \geq 0$ , consider the counting function

$$ \begin{align*} sq(\gamma_1,*,\mathcal{Q},L) := \#\left(\left\lbrace \begin{array}{@{}c | l@{}} \text{square-tiled surfaces }q & q \in \mathcal{Q}, \\ & \boldsymbol{\Re}(q) = \mathrm{Mod} \cdot \gamma_1, \\ & \mathrm{Area}(q) \leq L, \end{array} \right\rbrace \bigg/ \sim \right)\!, \end{align*} $$

where $\sim $ denotes the equivalence relation induced by cut and paste operations.

Using standard lattice point counting arguments in place of Theorem 3.2 in the proof of Theorem 1.1 yields the following result.

Theorem 3.9. There exists a constant $\kappa = \kappa (\mathcal {Q})> 0$ such that for every $L \geq 0$ ,

$$ \begin{align*} sq(\gamma_1,*,\mathcal{Q},L) = v(\gamma_1,\mathcal{Q}) \cdot L^h+ O_{\gamma_1,\mathcal{Q}}(L^{h-\kappa}). \end{align*} $$