Uniform distribution of saddle connection lengths in all $\textsf{SL}(2,\pmb {\mathbb {R}})$ orbits

Abstract

For every flat surface, almost every flat surface in its $\textsf{SL}(2,\mathbb {R})$ orbit has the following property: the sequence of its saddle connection lengths in non-decreasing order is uniformly distributed in the unit interval.

1 Introduction

Given a closed Riemann surface X of genus at least two, every non-zero holomorphic one-form $\omega$ on X has at least one zero. If $\Sigma$ is the set of zeroes of such an $\omega$ then on $X {\setminus } \Sigma$ there is an induced atlas of charts to $\mathbb {R}^2$ all of whose transition maps are translations. A saddle connection of a holomorphic one-form $\omega$ is any continuous map $v: [0,1] \rightarrow X$ such that $v^{-1}(\Sigma ) = \{0,1\}$ and that v|(0, 1) is a geodesic segment in every chart of the induced atlas on $X \setminus \Sigma$. Associated to each saddle connection v is its holonomy vector

$$\begin{aligned} \textsf{h}(v) = \left( \int \limits _v \textsf{Re}(\omega ),\int \limits _v \textsf{Im}(\omega ) \right) \end{aligned}$$

in $\mathbb {R}^2$. The group $\textsf{SL}(2,\mathbb {R})$ is known to act on pairs $(X,\omega )$ via composition with the charts of the atlas. The action takes saddle connections to saddle connections, and is such that $\textsf{h}(\textsf{g}v) = \textsf{g}\textsf{h}(v)$ for all $\textsf{g}\in \textsf{SL}(2,\mathbb {R})$ and all saddle connections of $\omega$.

Enumerating the saddle connections of $\omega$ as $n \mapsto v_n$ in such a way that $n \mapsto |\!|\textsf{h}(v_n) |\!|$ is non-decreasing one can ask about the distribution of this sequence modulo one. In [1] it was proved for every $\textsf{SL}(2,\mathbb {R})$ invariant and $\textsf{SL}(2,\mathbb {R})$ ergodic probability measure $\mu$ on any stratum ${\mathcal {H}}$, that the sequence $n \mapsto v_n$ is uniformly distributed modulo one for $\mu$ almost every flat surface. Here we prove the following refined result.

Theorem 1.1

For every flat surface $\omega _0$ of genus at least two and for almost every $\textsf{g}\in \textsf{SL}(2,\mathbb {R})$, the sequence $n \mapsto v_n$ of lengths of saddle connections of $\textsf{g}\omega _0$ listed in non-decreasing order is uniformly distributed modulo one.

This is an improvement over [1, Theorem 1] in two ways. Firstly, the conclusion is stronger as the result holds for Haar almost-every flat surface in every $\textsf{SL}(2,\mathbb {R})$ orbit, not just for almost every flat surface with respect to any ergodic $\textsf{SL}(2,\mathbb {R})$ invariant probability measure on a stratum. Secondly, the use of [6] and the reliance on spectral gap results for the action of $\textsf{SL}(2,\mathbb {R})$ on a stratum found in the proof of [1, Theorem 1] are replaced with a more direct argument that only makes use of the following quadratic upper bound on the number of saddle connections in a sector extending [2]. The notation is explained after the statement of the theorem.

Theorem 1.2

Fix a stratum ${\mathcal {H}}$ of flat surfaces of genus at least two. There is a constant $c_4$ with the following property. For every $\epsilon > 0$ and every flat surface $\omega \in {\mathcal {H}}$ there is $C(\omega ,\epsilon ) > 0$ such that

$$\begin{aligned} R \ge \dfrac{C(\omega ,\epsilon )}{|I|^{2-\epsilon }} \Rightarrow |\Lambda (\omega ;R) \cap \textsf{Sec}(I)| \le c_4 |I| R^2 \end{aligned}$$

(1.3)

for every arc $I \subset \textsf{S}^1$. Moreover, for each $\epsilon > 0$ the constant $C(\omega ,\epsilon )$ depends continuously on $\omega$.

The explicit description in (1.3) of how large R must be in terms of the arc length of I is of independent interest. Theorem 1.2 will be proved in “Appendix A”.

We now introduce some notation that will be used throughout the article. Given an arc $I \subset \textsf{S}^1$ write $\textsf{Sec}(I)$ for the subset of $\mathbb {R}^2$ that projects radially to I. Given $\alpha < \beta$ with $\beta - \alpha < 2\pi$ write $\textsf{Sec}(\alpha ,\beta )$ for the sector $\{ u \in \mathbb {R}^2: \alpha \le \arg (u) < \beta \}$. Given $0 \le A < B$ write $\textsf{Ann}(A,B)$ for the annulus $\{ u \in \mathbb {R}^2: A \le |\!|u |\!|\le B \}$. Fix a non-zero holomorphic one-form $\omega$ on a closed Riemann surface X of genus at least two. Write $\Lambda (\omega )$ for the set of saddle connections of $\omega$. For $R > 0$ write $\Lambda (\omega ;R)$ for the set of saddle connections of length at most R on $\omega$. We enumerate $\Lambda (\omega )$ by $n \mapsto v_n$ so that $n \mapsto |\!|\textsf{h}(v_n) |\!|$ is non-decreasing and $v \mapsto \arg (\textsf{h}(v)) \in [0,2\pi )$ is non-decreasing on each level set of $n \mapsto |\!|\textsf{h}(v_n) |\!|$. Given $N \in \mathbb {N}$ write $\Xi (\omega ;N)$ for $\{ v_n: n \le N \}$ where $n \mapsto v_n$ is the same enumeration as before. Lastly, write $\ell (\omega ;N) = |\!|\textsf{h}(v_N) |\!|$ for the length of the Nth saddle connection on $\omega$ in terms of the above ordering, and $\ell (\omega ) = \ell (\omega ;1)$ for the length of the shortest saddle connection on $\omega$. By an abuse of notation, given $B \subset \mathbb {R}^2$ we write $\Lambda (\omega ) \cap B$ for the set of saddle connections of $\omega$ whose holomony vectors belong to B. We will interpret $\Lambda (\omega ;R) \cap B$ and $\Xi (\omega ;N) \cap B$ similarly.

We thank Jon Chaika for suggesting this project and many useful conversations related to it. We also thank the anonymous referee for a very thorough report. Donald Robertson was supported by NSF grant DMS 1703597.

2 Beginning the proof of Theorem 1.1

Here we begin the proof of Theorem 1.1 by reducing it to Theorem 2.8 via the Weyl criterion and a Borel–Cantelli argument. Fix throughout this section a unit-area flat surface $\omega _0$ of genus at least two. Masur’s work [4, 5] yields constants $c_1,c_2 > 0$ with

$$\begin{aligned} (c_1eR)^2 \le |\Lambda (\omega ;R)| \le \left( c_2 \frac{R}{e} \right) ^2 \end{aligned}$$

(2.1)

for all $R > 0$. (The unorthodox expression is for later convenience.)

Write $\mu$ for Haar measure on $\textsf{SL}(2,\mathbb {R})$ and $\textsf{m}$ for Lebesgue measure on $\mathbb {R}$. For each $t \in \mathbb {R}$ write

$$\begin{aligned} \textsf{a}^t = \begin{bmatrix} e^t &{} 0 \\ 0 &{} e^{-t} \end{bmatrix} \qquad \textsf{r}^t = \begin{bmatrix} \cos (t) &{} -\sin (t) \\ \sin (t) &{} \cos (t) \end{bmatrix} \end{aligned}$$

and define ${{\,\mathrm{\textsf{D}}\,}}(T) = \{ \textsf{r}^\theta \textsf{a}^t \textsf{r}^\psi \in \textsf{SL}(2,\mathbb {R}): 0 \le \theta , \psi < 2 \pi , 0 \le t \le T \}$ for all $T > 0$. We scale $\mu$ once and for all such that $\mu ({{\,\mathrm{\textsf{D}}\,}}(2)) = 1$. Write $\chi _p(x) = \exp (2 \pi i p x)$ for each $p \in \mathbb {Z}$ and all $x \in \mathbb {R}$.

Our first reduction is to the following uniform distribution criterion. Although we prove the theorem for all $\tau > 1$, it suffices for the proof of Theorem 1.1 to do so for a sequence of values $\tau > 1$ converging to 1.

Theorem 2.2

Fix $p \in \mathbb {Z}$ and $\tau > 1$. One has

$$\begin{aligned} \lim _{J \rightarrow \infty } \frac{1}{\lceil \tau ^J \rceil } \sum _{v \in \Xi (\textsf{g}\omega _0; \lceil \tau ^J \rceil )} \chi _p(|\!|\textsf{h}(v) |\!|) = 0 \end{aligned}$$

(2.3)

for $\mu$ almost-every $\textsf{g}\in {{\,\mathrm{\textsf{D}}\,}}(1)$.

Proof of Theorem 1.1 assuming Theorem 2.2

Using Weyl’s criterion for uniform distribution and [1, Lemma 5] it suffices for the proof of Theorem 1.1 to produce a sequence $\tau _1> \tau _2 > \cdots \rightarrow 1$ with the property that

$$\begin{aligned} \lim _{J \rightarrow \infty } \frac{1}{\lceil \tau _i^J \rceil } \sum _{v \in \Xi (\textsf{g}\omega _0; \lceil \tau _i^J \rceil )} \chi _p(|\!|\textsf{h}(v) |\!|) = 0 \end{aligned}$$

(2.4)

holds for every $i,p \in \mathbb {N}$ and $\mu$ almost-every $\textsf{g}\in \textsf{SL}(2,\mathbb {R})$. Since the $\textsf{SL}(2,\mathbb {R})$ orbit of $\omega _0$ is countably covered by sets of the form ${{\,\mathrm{\textsf{D}}\,}}(1) \textsf{g}\omega _0$ it suffices to work over ${{\,\mathrm{\textsf{D}}\,}}(1)$. $\square$

Our next reduction will be via a Borel–Cantelli argument similar to the one in [1]. We give full details as there are several salient changes; most notably we average over $\Xi (\textsf{g}\omega _0; N)$ and ${{\,\mathrm{\textsf{D}}\,}}(1)$ in place of $\Lambda (\omega ; N^2)$ and a large compact subset of a stratum. For each $N \in \mathbb {N}$ define

$$\begin{aligned} f_N(\textsf{g}) = \dfrac{1}{N} \sum _{v \in \Xi (\textsf{g}\omega _0; N)} \chi _p(\Vert \textsf{h}(v) \Vert ) \end{aligned}$$

for all $\textsf{g}\in \textsf{SL}(2,\mathbb {R})$.

Lemma 2.5

For every $N \in \mathbb {N}$, every $0 < S \le 1$ and every $\sigma > 0$ the estimate

$$\begin{aligned}{} & {} \mu (\{ \textsf{g}\in {{\,\mathrm{\textsf{D}}\,}}(1): |f_N(\textsf{g})|^2 \ge N^{-\sigma } \} )^2 \nonumber \\{} & {} \quad \le 4 \int \limits _{{{\,\mathrm{\textsf{D}}\,}}(1)} \dfrac{\mu ({{\,\mathrm{\textsf{D}}\,}}(S) \textsf{x}\cap \{ \textsf{g}\in {{\,\mathrm{\textsf{D}}\,}}(1): |f_N(\textsf{g})|^2 \ge N^{-\sigma } \})}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \,\textrm{d}\mu (\textsf{x}) \end{aligned}$$

(2.6)

holds.

Proof

Fix $N \in \mathbb {N}$ and $0 < S \le 1$ and $\sigma > 0$. Write $M = \{ \textsf{g}\in {{\,\mathrm{\textsf{D}}\,}}(1): |f_N(\textsf{g})|^2 \ge N^{-\sigma } \}$. Consider the function $h: \textsf{SL}(2,\mathbb {R}) \rightarrow [0,1]$ defined by

$$\begin{aligned} h(\textsf{x}) = \dfrac{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2) \textsf{x}\cap M)}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \end{aligned}$$

for all $\textsf{x}\in \textsf{SL}(2,\mathbb {R})$.

Claim. $\mu \left( \left\{ \textsf{x}\in M: \dfrac{\mu ({{\,\mathrm{\textsf{D}}\,}}(S) \textsf{x}\cap M)}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \ge \dfrac{\mu (M)}{2} \right\} \right) \ge \int _{2h \ge \mu (M)} h \,\textrm{d}\mu$

Proof

If we have $\textsf{y}\in \textsf{SL}(2,\mathbb {R})$ with $h(\textsf{y}) \ge \mu (M) / 2$ and $\textsf{x}\in D(S/2) \textsf{y}$ then

$$\begin{aligned} \dfrac{\mu ({{\,\mathrm{\textsf{D}}\,}}(S) \textsf{x}\cap M)}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \ge \dfrac{\mu (M)}{2} \end{aligned}$$

because in this case ${{\,\mathrm{\textsf{D}}\,}}(S) \textsf{x}\cap M \supset {{\,\mathrm{\textsf{D}}\,}}(S/2) \textsf{y}\cap M$. Thus

$$\begin{aligned} \bigcup _{2h(\textsf{y}) \ge \mu (M)} {{\,\mathrm{\textsf{D}}\,}}(S/2) \textsf{y}\cap M \le \left\{ \textsf{x}\in M: \dfrac{\mu ({{\,\mathrm{\textsf{D}}\,}}(S) \textsf{x}\cap M)}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \ge \dfrac{\mu (M)}{2} \right\} \end{aligned}$$

and, combined with

$$\begin{aligned}&\mu \left( \bigcup _{2h(\textsf{y}) \ge \mu (M)} {{\,\mathrm{\textsf{D}}\,}}(S/2) \textsf{y}\cap M \right) \\&\quad = \dfrac{1}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \int \limits _{\textsf{SL}(2,\mathbb {R})} \mu \left( {{\,\mathrm{\textsf{D}}\,}}(S/2) \textsf{x}\cap \bigcup _{2h(\textsf{y}) \ge \mu (M)} {{\,\mathrm{\textsf{D}}\,}}(S/2) \textsf{y}\cap M \right) \,\textrm{d}\mu (\textsf{x})\\&\quad \ge \dfrac{1}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \int \limits _{2h \ge \mu (M)} \mu \left( {{\,\mathrm{\textsf{D}}\,}}(S/2) \textsf{x}\cap \bigcup _{2h(\textsf{y}) \ge \mu (M)} {{\,\mathrm{\textsf{D}}\,}}(S/2) \textsf{y}\cap M \right) \,\textrm{d}\mu (\textsf{x})\\&\quad \ge \dfrac{1}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \int \limits _{2h \ge \mu (M)} \mu \left( {{\,\mathrm{\textsf{D}}\,}}(S/2) \textsf{x}\cap M \right) \,\textrm{d}\mu (\textsf{x}) \end{aligned}$$

we have proved the claim. $\square$

Since $M \subset {{\,\mathrm{\textsf{D}}\,}}(1)$ we have $h = 0$ outside ${{\,\mathrm{\textsf{D}}\,}}(2)$. Thus

$$\begin{aligned} \int \limits _{2h < \mu (M)} h \,\textrm{d}\mu \le \dfrac{\mu (M)}{2} \mu ({{\,\mathrm{\textsf{D}}\,}}(2)) \end{aligned}$$

and we deduce

$$\begin{aligned} \dfrac{\mu (M)}{2} \le \left( 1 - \dfrac{\mu ({{\,\mathrm{\textsf{D}}\,}}(2))}{2} \right) \mu (M) \le \mu \left( \left\{ \textsf{x}\in M: \dfrac{\mu ({{\,\mathrm{\textsf{D}}\,}}(S) \textsf{x}\cap M)}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \ge \dfrac{\mu (M)}{2} \right\} \right) \end{aligned}$$

from the claim and our scaling of $\mu$ because the integral of h is $\mu (M)$ by Fubini. An application of Markov’s inequality then furnishes (2.6). $\square$

Proposition 2.7

There is $C_1 > 0$ such that for every $N \in \mathbb {N}$, every $0 < S \le 1$ and every $\sigma > 0$ we have

$$\begin{aligned}&\mu \left( \left\{ \textsf{g}\in {{\,\mathrm{\textsf{D}}\,}}(1) : |f_N(\textsf{g})|^2 > \dfrac{1}{N^\sigma } \right\} \right) \\&\quad \le \left( C_1 N^\sigma \dfrac{1}{2 \pi } \int \limits _0^{2 \pi } \int \limits _0^1 \dfrac{1}{2\pi } \int \limits _0^{2\pi } \dfrac{1}{S} \int \limits _0^S \dfrac{1}{N^2} \sum _{v,w \in \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0;N)} \chi _p(|\!|\textsf{h}(v) |\!|) \, \overline{\chi _p(|\!|\textsf{h}(w) |\!|)} \,\textrm{d}s \,\textrm{d}\theta \,\textrm{d}t \,\textrm{d}\phi \right) ^{1/2} \end{aligned}$$

Proof

Fix $N \in \mathbb {N}$ and $0 < S \le 1$ and $\sigma > 0$. Write $M = \{ \textsf{g}\in {{\,\mathrm{\textsf{D}}\,}}(1): |f_N(\textsf{g})|^2 \ge N^{-\sigma } \}$. We have

$$\begin{aligned} \mu (M)^2 \le 4 \int \limits _{{{\,\mathrm{\textsf{D}}\,}}(1)} \dfrac{\mu ({{\,\mathrm{\textsf{D}}\,}}(S) \textsf{x}\cap M)}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \,\textrm{d}\mu (x) = 4 \int \limits _{{{\,\mathrm{\textsf{D}}\,}}(1)} \dfrac{1}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \int \limits _{{{\,\mathrm{\textsf{D}}\,}}(S)} 1_M(\textsf{g}\textsf{x}) \,\textrm{d}\mu (\textsf{g}) \,\textrm{d}\mu (\textsf{x}) \end{aligned}$$

from Lemma 2.5. The definition of M along with Markov’s inequality gives

$$\begin{aligned} \mu (M)^2 \le 4 N^\sigma \int \limits _{{{\,\mathrm{\textsf{D}}\,}}(1)} \dfrac{1}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \int \limits _{{{\,\mathrm{\textsf{D}}\,}}(S)} |f_N( \textsf{g}\textsf{x}) |^2 \,\textrm{d}\mu (\textsf{g}) \,\textrm{d}\mu (\textsf{x}) \end{aligned}$$

and the right-hand integral becomes

$$\begin{aligned}{} & {} \dfrac{1}{2\pi } \int \limits _0^{2\pi } \dfrac{1}{2\pi } \int \limits _0^{2\pi } \int \limits _0^1 \dfrac{1}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \dfrac{1}{2\pi } \int \limits _0^{2\pi } \dfrac{1}{2\pi }\\{} & {} \quad \int \limits _0^{2\pi } \int \limits _0^S \sinh (t) \sinh (s) |f_N( \textsf{r}^\varphi \textsf{a}^s \textsf{r}^{\theta _1} \textsf{r}^{\theta _2} \textsf{a}^t \textsf{r}^\phi ) |^2 \,\textrm{d}s \,\textrm{d}\varphi \,\textrm{d}\theta _1 \,\textrm{d}t \,\textrm{d}\theta _2 \,\textrm{d}\phi \end{aligned}$$

upon writing $\mu$ in terms of the Cartan decomposition of $\textsf{SL}(2,\mathbb {R})$ as in [3, Proposition 5.28]. The rotation $\textsf{r}^\varphi$ does not affect the sum and may be removed; the integrals over $\textsf{r}^{\theta _1}$ and $\textsf{r}^{\theta _2}$ together form a convolution and may be combined into a single term; we may bound $\sinh (t)$ by 2 and $\sinh (s)$ by $\sinh (S)$. For some constant $C_1'$ one has

$$\begin{aligned} \dfrac{\sinh (S)}{\mu ({{\,\mathrm{\textsf{D}}\,}}(S/2))} \sim \dfrac{C_1'}{S} \end{aligned}$$

and, together with the above, we have an absolute constant $C_1 > 0$ with

$$\begin{aligned} \mu (M)^2\le & {} C_1 N^\sigma \dfrac{1}{2\pi } \int \limits _0^{2\pi } \int \limits _0^1 \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S}\\{} & {} \int \limits _0^{S} \frac{1}{N^2} \sum _{v,w \in \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0;N)} \chi _p(|\!|\textsf{h}(v) |\!|) \, \overline{\chi _p(|\!|\textsf{h}(w) |\!|)} \,\textrm{d}s \,\textrm{d}\theta \,\textrm{d}t \,\textrm{d}\phi \end{aligned}$$

as desired. $\square$

From now on we fix the relationship

$$\begin{aligned} S = \dfrac{1}{N^\delta } \end{aligned}$$

between $N \in \mathbb {N}$ and $S > 0$, where $\delta > 0$ is to be determined by future requirements. (Ultimately $\delta = \tfrac{1}{3}$ will suffice.) We now reduce the proof of Theorem 1.1 to the following statement.

Theorem 2.8

There is $\eta > 0$ and $N_0 \in \mathbb {N}$ and a constant $C > 0$ such that

$$\begin{aligned}{} & {} \dfrac{1}{2 \pi } \int \limits _0^{2 \pi } \int \limits _0^1 \dfrac{1}{2\pi } \int \limits _0^{2\pi } \dfrac{1}{S} \int \limits _0^S \dfrac{1}{N^2} \sum _{v,w \in \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0;N)} \chi _p(|\!|\textsf{h}(v) |\!|) \, \overline{\chi _p(|\!|\textsf{h}(w) |\!|)} \,\textrm{d}s \,\textrm{d}\theta \,\textrm{d}t \,\textrm{d}\phi \nonumber \\{} & {} \quad \le \dfrac{C}{N^\eta } \end{aligned}$$

(2.9)

holds for all $N \ge N_0$.

Proof of Theorem 1.1 assuming Theorem 2.8

Fix $p \in \mathbb {Z}$ and $\tau > 0$. By Theorem 2.2 it suffices to verify (2.3) for $\mu$ almost-every $\textsf{g}\in {{\,\mathrm{\textsf{D}}\,}}(1)$. Let $\eta > 0$ and $N_0 \in \mathbb {N}$ and $C > 0$ be as in the hypothesis. Fix $0< \sigma < \eta$. Whenever $\lceil \tau ^J \rceil \ge N_0$ and $\sigma > 0$ we have

$$\begin{aligned} \mu \left( \left\{ \textsf{g}\in {{\,\mathrm{\textsf{D}}\,}}(1): \left| \frac{1}{\lceil \tau ^J \rceil } \sum _{v \in \Xi (\textsf{g}\omega _0;\lceil \tau ^J \rceil )} \chi _p(|\!|\textsf{h}(v) |\!|) \right| ^2 \ge \frac{1}{\lceil \tau ^J \rceil ^\sigma } \right\} \right) \le \left( C_1 C \dfrac{\lceil \tau ^J \rceil ^\sigma }{\lceil \tau ^J \rceil ^\eta } \right) ^{1/2} \end{aligned}$$

by applying Proposition 2.7 and then (2.9). The right-hand side is summable over $J \in \mathbb {N}$ and the Borel–Cantelli lemma finishes the proof. $\square$

There are two major steps in the proof of Theorem 2.8. We outline them here and carry out the details in Sects. 3 and 4 respectively. The steps will be combined to prove Theorem 1.1 in Sect. 5.

Step 1: Annular estimate. We wish to move the action $\textsf{a}^s$ inside the summation appearing in (2.9). This is not straightforward because $\Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0; N)$ and $\textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0;N)$ need not agree. Indeed, if for $\textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0$ one knows $v_N$ is close to the horizontal and $v_{N + 1}$ is close to the vertical then it may be that $\textsf{a}^s v_{N + 1}$ is shorter than $\textsf{a}^s v_N$. To get around this issue it would suffice to find $\zeta > 0$ such that

$$\begin{aligned} \left| \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0;N) \triangle \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0;N) \right| \ll N^{1 - \zeta } \end{aligned}$$

holds for all $s,\theta ,t,\phi$. One can prove such an estimate using the effective count [6] for the number of saddle connections of length at most R as $R \rightarrow \infty$ but our goal is to avoid the use of spectral gap results. As a replacement we will find constants $\zeta > 0$ and $\lambda > 0$ such that

$$\begin{aligned} \mu \left( \left\{ 0 \le t \le 1: \begin{gathered} | \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0;N) \triangle \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0;N) | > N^{1 - \zeta } \\ for some (s,\theta ) \in [0,S] \times [0,2\pi ) \end{gathered} \right\} \right) \ll \dfrac{1}{N^\lambda } \end{aligned}$$

holds for all $0 \le \phi < 2\pi$. We will do so in Sect. 3 using Theorem 1.2.

Although $\Xi (\textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0; N)$ and $\textsf{r}^\theta \Xi (\textsf{a}^t \textsf{r}^\phi \omega _0; N)$ may also disagree as sets of saddle connections (because we have decided to order saddle connections of the same length by increasing angle) in this case the summations over the two sets agree because the summands only depend on the lengths of the saddle connections. It is therefore no problem, upon moving $\textsf{a}^s$ inside in (2.9), to move the action $\textsf{r}^\theta$ inside as well.

The purpose of the annular estimate is to reduce the verification of (2.9) to the production of some $\eta > 0$ such that

$$\begin{aligned}{} & {} \int \limits _0^1 \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \frac{1}{N^2} \sum _{v,w \in \Xi (\textsf{a}^t \textsf{r}^\phi \omega _0;N)} \chi _p(|\!|\textsf{a}^s \textsf{r}^\theta \textsf{h}(v) |\!|) \, \overline{\chi _p(|\!|\textsf{a}^s \textsf{r}^\theta \textsf{h}(w) |\!|)} \,\textrm{d}s \,\textrm{d}\theta \,\textrm{d}t\nonumber \\{} & {} \quad \ll \frac{1}{N^\eta } \end{aligned}$$

(2.10)

for every $0 \le \phi < 2 \pi$.

Step 2: Controlling pairs. To produce $\eta > 0$ such that (2.10) holds, we apply a linearization to arrive at the quantity

$$\begin{aligned} \int \limits _0^1 \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \frac{1}{N^2} \sum _{v,w \in \Xi (\textsf{a}^t \textsf{r}^\phi \omega _0;N)} \chi _p(s \alpha (\textsf{r}^\theta \textsf{h}(v))) \, \overline{\chi _p(s \alpha (\textsf{r}^\theta \textsf{h}(w)))} \,\textrm{d}s \,\textrm{d}\theta \,\textrm{d}t \end{aligned}$$

which we need to control for every $0 \le \phi < 2 \pi$. From the proof of [1, Lemma 12] it suffice to bound

$$\begin{aligned} |\!|\textsf{h}(v) |\!|| \sin (2 \theta _{v,w}) | \end{aligned}$$

from below by a power of N. Here $\theta _{v,w}$ is the angle between the holonomy vectors of the saddle connections v and w. This issue will be dealt with in Sect. 4.

3 Annular estimate

In this section we will establish the following theorem.

Theorem 3.1

There are $\lambda > 0$ and $\zeta > 0$ and $N_1 > 0$ such that

$$\begin{aligned} \textsf{m}\left( \left\{ 0 \le t \le 1: \begin{gathered} | \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0; N) \triangle \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0; N)| > N^{1 - \zeta } \\ for some (s,\theta ) \in [0,S] \times [0,2\pi ) \end{gathered} \right\} \right) \ll \frac{1}{N^\lambda } \end{aligned}$$

holds for all $N \ge N_1$ and all $0 \le \phi < 2\pi$.

The proof of Theorem 3.1 will take up the remainder of this section. Recall that $\delta > 0$ defines $S = N^{-\delta }$. Below, all requirements that N be large enough depend only on $\omega _0$ and not on $\omega$. Throughout this section fix $0 \le \phi < 2 \pi$ and write $\omega = \textsf{r}^\phi \omega _0$.

We begin with two lemmas that will be used to relate

$$\begin{aligned} | \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \omega ; N) \triangle \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \omega ; N)| > N^{1 - \zeta } \end{aligned}$$

with counts for saddle connections in certain annuli.

Lemma 3.2

For every $\textsf{g}\in \textsf{SL}(2,\mathbb {R})$ one has

$$\begin{aligned} \Xi (\textsf{a}^s \textsf{g}\omega ;N) \subset \textsf{a}^s \Lambda (\textsf{g}\omega ;e^{2s} \ell (\textsf{g}\omega ;N)) \end{aligned}$$

for every $s > 0$ and every $N \in \mathbb {N}$.

Proof

If a saddle connection v of $\textsf{g}\omega$ has a length of more than $e^{2s} \ell (\textsf{g}\omega ;N)$ then $\textsf{a}^s v$ has a length of more than $e^s \ell (\textsf{g}\omega ;N)$. The saddle connection $\textsf{a}^s v$ therefore cannot be amongst the first N saddle connections of $\textsf{a}^s \textsf{g}\omega$ since $\textsf{a}^s \Xi (\textsf{g}\omega ;N)$ has cardinality exactly N and all of its members are saddle connections of $\textsf{a}^s \textsf{g}\omega$ of length at most $e^s \ell (\textsf{g}\omega ;N)$. $\square$

Lemma 3.3

For every $\textsf{g}\in \textsf{SL}(2,\mathbb {R})$ one has

$$\begin{aligned} \Xi (\textsf{a}^s \textsf{g}\omega ; N) \supset \textsf{a}^s \Lambda (\textsf{g}\omega ; e^{-2s} \ell (\textsf{g}\omega ;N)) \end{aligned}$$

for every $s > 0$ and every $N \in \mathbb {N}$.

Proof

Every saddle connection in $\textsf{a}^s \Lambda (\textsf{g}\omega ; e^{-2s} \ell (\textsf{g}\omega ;N))$ is a saddle connection of $\textsf{a}^s \textsf{g}\omega$ with length at most $e^{-s} \ell (\textsf{g}\omega ;N)$. Moreover, no saddle connection of $\textsf{g}\omega$ with length greater than $\ell (\textsf{g}\omega ;N)$ will, under the image of $\textsf{a}^s$, be shorter than any saddle connection of $\textsf{a}^s \Lambda (\textsf{g}\omega ;e^{-2s} \ell (\textsf{g}\omega ;N))$. So all saddle connections in $\textsf{a}^s \Lambda (\textsf{g}\omega ;e^{-2s} \ell (\textsf{g}\omega ;N))$ are amongst the first N saddle connections of $\textsf{a}^s \textsf{g}\omega$. $\square$

If the set

$$\begin{aligned} \{ 0 \le t \le 1: | \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \omega ; N) \triangle \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \omega ; N)| > N^{1 - \zeta } for some (s,\theta ) \in [0,S] \times [0,2\pi ) \}\nonumber \\ \end{aligned}$$

(3.4)

is empty for some $N \in \mathbb {N}$ then there is nothing to prove for that N. Suppose that t belongs to (3.4) for some $N \in \mathbb {N}$. We get $0 \le s \le S$ and $0 \le \theta < 2\pi$ depending on t such that

$$\begin{aligned} \left| \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \textsf{r}^\psi \omega ; N) \triangle \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \textsf{r}^\psi \omega ; N) \right| \ge N^{1 - \zeta } \end{aligned}$$

(3.5)

holds. Given (3.5) we estimate

$$\begin{aligned}&\left| \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \omega ; N) \triangle \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \omega ; N) \right| \\&\quad \le \left| \textsf{a}^s \Lambda (\textsf{r}^\theta \textsf{a}^t \omega ; e^{2s} \ell (\textsf{r}^\theta \textsf{a}^t \omega ; N)) \setminus \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \omega ; N) \right| \\&\qquad + \left| \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \omega ; N) \setminus \textsf{a}^s \Lambda (\textsf{r}^\theta \textsf{a}^t \omega ; e^{-2s} \ell (\textsf{r}^\theta \textsf{a}^t \omega ;N)) \right| \\&\quad \le \left| \Lambda (\textsf{r}^\theta \textsf{a}^t \omega ) \cap \textsf{Ann}(e^{-2S} \ell (\textsf{r}^\theta \textsf{a}^t \omega ; N) , e^{2S} \ell (\textsf{r}^\theta \textsf{a}^t \omega ; N)) \right| \end{aligned}$$

from Lemmas 3.2 and 3.3, where we have used $s \le S$ in deducing the last inequality. Thus, whenever t belongs to (3.4) for some $N \in \mathbb {N}$, we have

$$\begin{aligned} N^{1 - \zeta }&\le \left| \Lambda (\textsf{r}^\theta \textsf{a}^t \omega ) \cap \textsf{Ann}(e^{-2S} \ell (\textsf{r}^\theta \textsf{a}^t \omega ; N) , e^{2S} \ell (\textsf{r}^\theta \textsf{a}^t \omega ; N)) \right| \\&= \left| \Lambda (\textsf{a}^t \omega ) \cap \textsf{Ann}(e^{-2S} \ell (\textsf{a}^t \omega ; N) , e^{2S} \ell (\textsf{a}^t \omega ; N)) \right| \end{aligned}$$

because $\Lambda (\textsf{r}^\theta \textsf{a}^t \omega ) = \textsf{r}^\theta \Lambda (\textsf{a}^t \omega )$ and $\textsf{r}^\theta$ does not change the length of the Nth saddle connection. We are led to consider the quantity

$$\begin{aligned} E_N(v;t) = {\left\{ \begin{array}{ll} 1 &{} \textsf{a}^t v \in \textsf{Ann}(e^{-2S} \ell (\textsf{a}^t \omega ; N), e^{2S} \ell (\textsf{a}^t \omega ; N)) \\ 0 &{} otherwise \end{array}\right. } \end{aligned}$$

(3.6)

defined for all $0 \le t \le 1$ and all $v \in \Lambda (\omega )$ and all $N \in \mathbb {N}$. By its definition we therefore have

$$\begin{aligned} N^{1-\zeta } \le \left| \Lambda (\textsf{a}^t \omega ) \cap \textsf{Ann}(e^{-2S} \ell (\textsf{a}^t \omega ; N), e^{2S} \ell (\textsf{a}^t \omega ; N)) \right| \le \sum _{v \in \Lambda (\omega )} E_N(v;t) \end{aligned}$$

whenever t belongs to (3.4). Thus

$$\begin{aligned} (3.4) \subset \left\{ 0 \le t \le 1: \sum _{v \in \Lambda (\omega )} E_N(v;t) \ge N^{1 - \zeta } \right\} \end{aligned}$$

which, together with Markov’s inequality, gives

$$\begin{aligned} \textsf{m}((3.4)) \le \frac{1}{N^{2(1-\zeta )}} \int \limits _0^1 \left| \sum _{v \in \Lambda (\omega )} E_N(v;t) \right| ^2 \,\textrm{d}\textsf{m}(t) \end{aligned}$$

(3.7)

so our focus now is to bound

$$\begin{aligned} \int \limits _0^1 \sum _{v \in \Lambda (\omega )} \sum _{w \in \Lambda (\omega )} E_N(v;t) E_N(w;t) \,\textrm{d}\textsf{m}(t) \end{aligned}$$

(3.8)

for N large. We begin with the following lemmas, which will allow us to restrict the sums in (3.8) to thin annuli.

Lemma 3.9

There are constants $c_1,c_2 > 0$ such that for all $0 \le t \le 1$ we have

$$\begin{aligned} \frac{1}{c_2} \frac{N^{0.5}}{\log N} \le \ell (\textsf{a}^t \omega ;N) \le \frac{1}{c_1} N^{0.5} \end{aligned}$$

for all $N \ge 2$.

Proof

First, note that

$$\begin{aligned} |\Lambda (\textsf{a}^t \omega ) \cap \textsf{B}(0,R)| = |\textsf{a}^t \Lambda (\omega ) \cap \textsf{B}(0,R)| = |\Lambda (\omega ) \cap \textsf{a}^{-t} \textsf{B}(0,R)| \end{aligned}$$

and $\textsf{B}(0,\frac{1}{e}R) \subset \textsf{a}^{-t} \textsf{B}(0,R) \subset \textsf{B}(0,eR)$ give from (2.1) the bounds $(c_1 R)^2 \le |\Lambda (\textsf{a}^t \omega ;R)| \le (c_2 R)^2$ for all $0 \le t \le 1$ and all $R > 0$. Taking $R = N^{0.5}/c_1$ gives $N \le |\Lambda (\textsf{a}^t \omega ; N^{0.5}/c_1)|$ whence $\ell (\textsf{a}^t \omega ;N) \le N^{0.5}/c_1$. Taking $R = N^{0.5}/c_2 \log N$ gives $N^{0.5}/ c_2\log N \le \ell (\textsf{a}^t \omega ;N)$. $\square$

Lemma 3.10

If $N^\delta \ge 2$ and $v \in \Lambda (\omega )$ is outside the annulus

$$\begin{aligned} \textsf{Ann}\left( \frac{1}{2ec_2} \frac{N^{0.5}}{\log N}, \frac{2e}{c_1} N^{0.5} \right) \end{aligned}$$

(3.11)

then $E_N(v;t) = 0$ for all $0 \le t \le 1$.

Proof

Fix $N \in \mathbb {N}$ with $N^\delta \ge 2$ and suppose $E_N(v;t) = 1$ for some $0 \le t \le 1$. By Lemma 3.9 we have

$$\begin{aligned} \frac{1}{e^{2S} c_2} \frac{N^{0.5}}{\log N} \le e^{-2S} \ell (\textsf{a}^t \omega ;N) \le e^{2S} \ell (\textsf{a}^t \omega ; N) \le \frac{e^{2S}}{c_1} N^{0.5} \end{aligned}$$

whence

$$\begin{aligned} \textsf{a}^t v \in \textsf{Ann}\left( \frac{1}{2 c_2} \frac{N^{0.5}}{\log N}, \frac{2}{c_1} N^{0.5} \right) \end{aligned}$$

because $e^{2S} \le 2$. Therefore

$$\begin{aligned} v \in \textsf{Ann}\left( \frac{1}{2ec_2} \frac{N^{0.5}}{\log N}, \frac{2e}{c_1} N^{0.5} \right) \end{aligned}$$

as $\textsf{a}^t$ can lengthen vectors by a facor of at most e and shorten them by a facor of at most 1/e. $\square$

Thus, for $N^\delta \ge 2$, only saddle connections of $\omega$ with holonomy inside (3.11) contribute to the summations in (3.8). We continue by partitioning the annulus (3.11) into sectors as follows. First define

$$\begin{aligned} {\overline{Z}} = \textsf{Sec}(-\psi ,\psi ) \cup \textsf{Sec}(\tfrac{\pi }{2} - \psi , \tfrac{\pi }{2} + \psi ) \cup \textsf{Sec}(\pi - \psi , \pi + \psi ) \cup \textsf{Sec}(\tfrac{3\pi }{2} - \psi , \tfrac{3\pi }{2} + \psi ) \end{aligned}$$

and

$$\begin{aligned} Z = {\overline{Z}} \cap \Lambda \left( \omega ; \frac{2e}{c_1} N^{0.5} \right) \end{aligned}$$

where $\psi = 2N^{-\gamma }$ for some $\gamma > 0$ yet to be determined. (In fac $\gamma = \frac{1}{300}$ will suffice.) Put

$$\begin{aligned} \kappa = \left\lfloor (\tfrac{\pi }{2} - 2 \psi ) \ell (\omega ;N)^\alpha \right\rfloor \end{aligned}$$

(3.12)

for some $\alpha > 0$ yet to be determined. (In fac $\alpha = \frac{1}{100}$ will suffice.) Decompose

$$\begin{aligned} W = \Lambda (\omega ) \cap \textsf{Ann}\left( \frac{1}{2ec_2} \frac{N^{0.5}}{\log N}, \frac{2e}{c_1} N^{0.5} \right) \setminus Z \end{aligned}$$

into $4\kappa$ subsets $W(1),\ldots ,W(4\kappa )$ each obtained by intersecting W with annular arcs ${\overline{W}}(1),\ldots ,{\overline{W}}(4\kappa )$ of size $(\frac{\pi }{2} - 2\psi )/\kappa$. Given $1 \le k \le 4\kappa$ let ${\overline{V}}(k)$ be the union of ${\overline{W}}(k)$, its reflections in the other quadrants, and any ${\overline{W}}(i)$ that are adjacent to any of these reflections. (See Fig. 1 for a schematic.) Lastly, put $V(k) = {\overline{V}}(k) \cap \Lambda (\omega )$.

Fig. 1

The annular sectors ${\overline{W}}(1)$ and ${\overline{W}}(k)$ for some $1< k < \kappa$ are in white. The set ${\overline{V}}(k)$ consists of twelve annular sectors: the eleven light grey sectors together with ${\overline{W}}(k)$. The set ${\overline{Z}}$ is shown in dark grey. Note that ${\overline{V}}(1)$ only consists of eight regions as ${\overline{Z}}$ is not considered adjacent to any of our regions

By Lemma 3.10 the right-hand side of (3.8) is bounded by the sum of the following expressions.

• $\sum _{k=1}^{4\kappa } \sum _{v \in W(k)} \sum _{w \in V(k)} \int _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t$

• $\sum _{v \in Z} \sum _{w \in \Lambda (\omega )} \int _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t + \sum _{v \in \Lambda (\omega )} \sum _{w \in Z} \int _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t$

• $\sum _{k=1}^{4\kappa } \sum _{v \in W(k)} \sum _{\begin{array}{c} w \in W\\ w \notin V(k) \end{array}} \int _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t$

and our goal now is to obtain power bounds for each of them. This is carried out in the next two subsections: the first two expressions will be bounded via sectorial counts and the third via a separation argument. Both make use of Theorem 1.2.

3.1 Sectorial count

Our goal here is to bound the sums

• $\sum _{k=1}^{4\kappa } \sum _{v \in W(k)} \sum _{w \in V(k)} \int _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t$

• $\sum _{v \in Z} \sum _{w \in \Lambda (\omega )} \int _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t$

by powers of N.

Let $C(\omega ,\epsilon )$ be as in Theorem 1.2. The parameter $\epsilon > 0$ will be determined later. (In fac $\epsilon = \frac{1}{100}$ suffices.) Fix $1 \le k \le 4 \kappa$. With I the appropriate sector of length $3(\frac{\pi }{2} - 2 \psi )/\kappa$ we have

$$\begin{aligned} |\Lambda (\omega ) \cap V(k)| \le 4|\Lambda (\omega ;\tfrac{2e}{c_1} N^{0.5}) \cap \textsf{Sec}(I)| \le 4 \cdot c_4 \cdot \frac{3(\frac{\pi }{2} - 2\psi )}{\kappa } \cdot \left( \frac{2e}{c_1} N^{0.5} \right) ^2 \end{aligned}$$

whenever

$$\begin{aligned} \frac{2e}{c_1} N^{0.5} \ge C(\omega ,\epsilon ) \left( \frac{\kappa }{3(\frac{\pi }{2} - 2\psi )} \right) ^{2+\epsilon } \end{aligned}$$

(3.13)

holds.

A priori, the occurrence of $\omega$ in (3.13) means all subsequent statements requiring N to be large enough will depend on $\omega$. However, since $\epsilon > 0$ will be fixed and $\omega \mapsto C(\omega ,\epsilon )$ is then continuous, the relation $\omega = \textsf{r}^\psi \omega _0$ implies that the apparent dependence on $\omega$ is in fac only a dependence on $\omega _0$.

Applying Lemma 3.9 and $\frac{1}{2} x \le \lfloor {x} \rfloor \le x$ on $[1,\infty )$ to (3.12), we bound $\kappa$ by

$$\begin{aligned} \frac{\pi }{8 c_2^\alpha } \frac{N^{\frac{\alpha }{2}}}{(\log N)^\alpha } \le \frac{1}{2} \left( \frac{\pi }{2} - 2 \psi \right) \left( \frac{1}{c_2} \frac{N^{0.5}}{\log N} \right) ^\alpha \le \kappa \le \left( \frac{\pi }{2} - 2 \psi \right) \left( \frac{1}{c_1} N^{0.5} \right) ^\alpha \le \frac{\pi }{2 c_1^\alpha } N^{\frac{\alpha }{2}} \end{aligned}$$

whenever $N^\gamma \ge 16/\pi$. Thus there is an absolute constant $c_6 > 0$ such that

$$\begin{aligned} |\Lambda (\omega ) \cap V(k)| \le 4c_4 \cdot \frac{3 \pi }{2} \frac{8 c_2^\alpha }{\pi } \frac{(\log N)^\alpha }{N^\frac{\alpha }{2}} \cdot \frac{4e^2}{c_1^2} N \le c_6N^{1 - \frac{\alpha }{2}} (\log N)^{\alpha } \le c_6 N^{1 - \frac{\alpha }{4}} \end{aligned}$$

whenever

$$\begin{aligned} N^{0.5} \ge \frac{c_1}{2e} \cdot \left( \frac{2}{3\pi } \right) ^{2+\epsilon } \cdot \left( \frac{\pi }{8 c_2} \frac{N^{\frac{\alpha }{2}}}{(\log N)^\alpha } \right) ^{2 + \epsilon } \cdot C(\omega ,\epsilon ) \end{aligned}$$

holds, which will be the case for N large enough provided

$$\begin{aligned} \alpha (2 + \epsilon ) < 1 \end{aligned}$$

(R)

is in place. We can therefore say, using Lemma 3.10 and (2.1) to bound the first two sums, and bounding the integral by 1, that

$$\begin{aligned} \sum _{k=1}^{4\kappa } \sum _{v \in W(k)} \sum _{w \in V(k)} \int \limits _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t\ll & {} N^{1 - \frac{\alpha }{4}} \sum _{k=1}^{4\kappa } \sum _{v \in W(k)} \int \limits _0^1 E_N(v;t) \,\textrm{d}\textsf{m}(t) \nonumber \\\ll & {} N^{2 - \frac{\alpha }{4}} \end{aligned}$$

(3.14)

holds whenever N is large enough.

For the second sum we again apply Theorem 1.2, this time with $\epsilon ' > 0$ to be determined ($\epsilon ' = \frac{1}{100}$ suffices) and I an appropriate arc of size $2\psi$ giving a constant $C(\omega ,\epsilon ')$ such that

$$\begin{aligned} |\Lambda (\omega ) \cap Z| \le 4c_4 \cdot 4N^{-\gamma } \cdot \left( \frac{2e}{c_1} N^{0.5} \right) ^2 \end{aligned}$$

whenever

$$\begin{aligned} \frac{2e}{c_1} N^{0.5} \ge \left( \frac{N^\gamma }{4} \right) ^{2 + \epsilon '} C(\omega ,\epsilon ') \end{aligned}$$

holds. Thus

$$\begin{aligned} |\Lambda (\omega ) \cap Z| \ll N^{1 - \gamma } \end{aligned}$$

holds provided

$$\begin{aligned} \frac{1}{2} > \gamma (2 + \epsilon ') \end{aligned}$$

(R)

is the case. Using Lemma 3.10 and (2.1) to bound the second sum, we can say that

$$\begin{aligned} \sum _{v \in Z} \sum _{w \in \Lambda (\omega )} \int \limits _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t \ll N^{2 - \gamma } \end{aligned}$$

(3.15)

holds for N large enough.

3.2 Separation

In this subsection we control the sum

$$\begin{aligned} \sum _{k=1}^{4\kappa } \sum _{v \in W(k)} \sum _{\begin{array}{c} w \in W \\ w \notin V(k) \end{array}} \int \limits _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t \end{aligned}$$

by a power of N.

Fix $1 \le k \le 4\kappa$ and fix saddle connections v, w of $\omega$ with $v \in W(k)$ and $w \in W {\setminus } V(k)$. Since $E_N(u;t)$ is unchanged when u is reflected in either the horizontal or the vertical axis, we may assume that v and w are in the first quadrant. Write $\theta _u = \arg (u)$. We assume that $\theta _v > \theta _w$ as the alternative involves identical arguments. The following properties are immediate consequences of that assumption, $v \in W(k)$ and $w \in W {\setminus } V(k)$.

S1:

$\dfrac{1}{2ec_2} \dfrac{N^{0.5}}{\log N} \le |\!|v |\!|, |\!|w |\!|\le \dfrac{2e}{c_1} N^{0.5}$

S2:

$\dfrac{2}{N^\gamma } \le \theta _w < \theta _v \le \dfrac{\pi }{2} - \dfrac{2}{N^\gamma }$

S3:

$\theta _v - \theta _w > \dfrac{1}{\ell (\omega ;N)^\alpha }$

Indeed S1 follows from Lemma 3.10, S2 follows from the definitions of $\psi$ and W, and S3 follows from (3.12) and N being large enough.

With these properties to hand, our goal in this subsection is bounding the Lebesgue measure of the set

$$\begin{aligned} \{ 0 \le t \le 1: E_N(v;t) E_N(w;t) = 1 \} \end{aligned}$$

(3.16)

by a negative power of N.

Lemma 3.17

For all $t > 0$ the angle between $\textsf{a}^t v$ and $\textsf{a}^t w$ is at least $\frac{1}{e^{2t}} (\theta _v - \theta _w)$.

Proof

The Cauchy mean value theorem gives

$$\begin{aligned} \arctan \left( \frac{1}{e^{2t}} \frac{v_2}{v_1} \right) - \arctan \left( \frac{1}{e^{2t}} \frac{w_2}{w_1} \right)\ge & {} \frac{1}{e^{2t}} \left( \arctan \left( \frac{v_2}{v_1} \right) - \arctan \left( \frac{w_2}{w_1} \right) \right) \\ {}= & {} \frac{1}{e^{2t}} (\theta _v - \theta _w) \end{aligned}$$

as desired. $\square$

We frequently use the estimates

$$\begin{aligned} \frac{1}{e^s} |\!|u |\!|\le |\!|\textsf{a}^s u |\!|\le e^s |\!|u |\!|\quad \theta _{\textsf{a}^s u} \le \frac{1}{e^{2s}} \theta _u \end{aligned}$$

which hold for all $s \ge 0$ and all u in the first quadrant.

If $E_N(v;t) E_N(w;t) = 0$ for all $0 \le t \le 1$ there is no need for a bound as (3.16) will have zero measure. We therefore assume also that there is a time $0 \le r \le 1$ at which $E_N(v;r) E_N(w;r) = 1$. The definition of $E_N$ and Lemma 3.9 give

$$\begin{aligned} \Big | |\!|\textsf{a}^r v |\!|- |\!|\textsf{a}^r w |\!|\Big | \le 2 \ell (\textsf{a}^t \omega ;N) \sinh (2 S) \le \frac{8}{c_1} N^{0.5 - \delta } \end{aligned}$$

(3.18)

as $\sinh (S) \sim S$ as $N \rightarrow \infty$. Also S3 and Lemmas 3.9, 3.17 give

$$\begin{aligned} \theta _{\textsf{a}^r v} - \theta _{\textsf{a}^r w} \ge \frac{1}{e^{2r}} (\theta _v - \theta _w) \ge \frac{c_1^\alpha }{e^2} N^{-\frac{\alpha }{2}} \end{aligned}$$

(3.19)

and our first goal is to deduce from these that there is horizontal and vertical separation of $\textsf{a}^r v$ from $\textsf{a}^r w$. Write $\chi$ for the angular separation $\theta _{\textsf{a}^r v} - \theta _{\textsf{a}^r w}$ and $Q = | |\!|\textsf{a}^r v |\!|- |\!|\textsf{a}^r w |\!||$ for the difference in length.

Fig. 2

If $\textsf{a}^r w$ is separated in angle from $\textsf{a}^r v$ as in (3.19) but $| |\!|\textsf{a}^r v |\!|- |\!|\textsf{a}^r w |\!||$ is not too large as in (3.18) (so that $\textsf{a}^r w$ belongs to the grey region) then we can say something about the horizontal and vertical separation of $\textsf{a}^r v$ and $\textsf{a}^r w$

Lemma 3.20

There are constants $K > 0$ and $\xi > 0$ such that $(\textsf{a}^r w)_1 - (\textsf{a}^r v)_1 \ge K N^{0.5 - \xi }$ and $(\textsf{a}^r v)_2 - (\textsf{a}^r w)_2 \ge K N^{0.5 - \xi }$ for all N large enough.

Proof

The coordinate $(\textsf{a}^r w)_1$ cannot (cf. Fig. 2) be smaller than $(|\!|\textsf{a}^r v |\!|- Q) \cos (\theta _{\textsf{a}^r v} - \chi )$ so from

$$\begin{aligned} \cos (\theta _{\textsf{a}^r v} - \chi ) - \cos (\theta _{\textsf{a}^r v}) = \sin (\theta _{\textsf{a}^r v}) \sin (\chi ) - (1 - \cos (\chi )) \cos (\theta _{\textsf{a}^r v}) \ge \frac{\theta _{\textsf{a}^r v}}{2} \frac{\chi }{2} - \frac{\chi ^2}{2} \end{aligned}$$

we have by (3.18), (3.19) and Lemma 3.10 that

$$\begin{aligned} (\textsf{a}^r w)_1 - (\textsf{a}^r v)_1&\ge |\!|\textsf{a}^r v |\!|\frac{\chi }{2} \left( \frac{\theta _{\textsf{a}^r v}}{2} - \chi \right) - Q \\&\ge \frac{1}{2e^2 c_2} \frac{N^{0.5}}{\log N} \cdot \frac{c_1^\alpha }{2e^2} N^{-\frac{\alpha }{2}} \left( \frac{1}{e^2} N^{-\gamma } - \frac{c_1^\alpha }{e^2} N^{-\frac{\alpha }{2}} \right) - \frac{8}{c_1} N^{0.5 - \delta } \end{aligned}$$

because

$$\begin{aligned} \dfrac{\theta _{\textsf{a}^r v}}{2} - \chi \ge \frac{1}{2e^2} \theta _v - \chi \ge \frac{1}{e^2} \frac{1}{N^\gamma } - \frac{c_1^\alpha }{e^2} \frac{1}{N^{\frac{\alpha }{2}}} > 0 \end{aligned}$$

for N large. The above is contingent on the inequalities

$$\begin{aligned} \gamma + \alpha< \delta \qquad \qquad \gamma < \frac{\alpha }{2} \end{aligned}$$

(R)

but provided they are satisfied we conclude there is a constant $K_1 > 0$ such that

$$\begin{aligned} (\textsf{a}^r w)_1 - (\textsf{a}^r v)_1 \ge \frac{K_1}{\log N} N^{0.5 - (\gamma + \alpha )} \end{aligned}$$

for all N large enough.

Similarly, the largest $(\textsf{a}^r w)_2$ can be is $(|\!|\textsf{a}^r v |\!|+ Q) \sin (\theta _{\textsf{a}^r v} - \chi )$ so from

$$\begin{aligned} \sin (\theta _{\textsf{a}^r v}) - \sin (\theta _{\textsf{a}^r v} - \chi ) = (1 - \cos (\chi ))\sin (\theta _{\textsf{a}^r v}) + \cos (\theta _{\textsf{a}^r v}) \sin (\chi ) \ge \frac{\chi ^2}{8} \frac{\theta _{\textsf{a}^r v}}{2} \end{aligned}$$

we have

$$\begin{aligned} (\textsf{a}^r v)_2 - (\textsf{a}^r w)_2&\ge |\!|\textsf{a}^r v |\!|\frac{\chi ^2}{8} \frac{\theta _{\textsf{a}^r v}}{2} - Q \\&\ge \frac{1}{2e^2 c_2} \frac{N^{0.5}}{\log N} \cdot \frac{c_1^{2\alpha }}{8e^4} N^{-\alpha } \cdot \frac{1}{e^2} N^{-\gamma } - \frac{8}{c_1} N^{0.5 - \delta } \end{aligned}$$

as above. Provided

$$\begin{aligned} \gamma + \alpha < \delta \end{aligned}$$

(R)

we conclude that there is a constant $K_2 > 0$ such that

$$\begin{aligned} (\textsf{a}^r v)_2 - (\textsf{a}^r w)_2 \ge \frac{K_2}{\log N} N^{0.5 - (\gamma + \alpha )} \end{aligned}$$

for all N large enough.

To conclude take $K = \min \{ K_1, K_2 \}$ and $\xi = \gamma + 2 \alpha$ and $N_2 \ge \max \{ M_1, M_2 \}$ large enough. $\square$

Define $f_{v,w}(t) = |\!|\textsf{a}^t v |\!|- |\!|\textsf{a}^t w |\!|$ for $t \in \mathbb {R}$. Certainly $f_{v,w}$ is continuous. Controlling the size of the set of those $0 \le t \le 1$ where $|f_{v,w}(t)|$ is small will be enough to bound the Lebesgue measure of (3.16). From Lemma 3.20 and Facts B.4, B.5 in “Appendix B” we conclude that $f_{v,w}$ is decreasing and has a unique zero. Since $f_{v,w}$ is continuous and decreasing, (3.18) furnishes $0 \le r_0 \le 1$ minimal with $|f_{v,w}(r_0)| \le \frac{8}{c_1} N^{0.5 - \delta }$.

Lemma 3.21

We have

$$\begin{aligned} |f_{v,w}(r_0 + N^{-\varpi })| \ge \frac{8}{c_1} N^{0.5 - \delta } \end{aligned}$$

for some $\varpi > 0$ to be determined. (In fac, one can take $\varpi = \frac{1}{100}$.)

Proof

Suppose that the contrary holds. Then

$$\begin{aligned} \left| |\!|\textsf{a}^{r_0 + t} v |\!|- |\!|\textsf{a}^{r_0 + t} w |\!|\right| = |f_{v,w}(r_0 + t)| \le \frac{8}{c_1} N^{0.5 - \delta } \end{aligned}$$

for all $0 \le t \le N^{-\varpi }$. We also have

$$\begin{aligned} \frac{|\cos (2 \theta _{\textsf{a}^{r_0 + t} v}) - \cos (2 \theta _{\textsf{a}^{r_0 + t} w})|}{|2 \theta _{\textsf{a}^{r_0 + t} v} - 2 \theta _{\textsf{a}^{r_0 + t} w}|} = \sin (2 \xi ) \ge \min \{ \sin (2 \theta _{\textsf{a}^{r_0 + t} w}), \sin (2 \theta _{\textsf{a}^{r_0 + t} v}) \} \end{aligned}$$

for some $\theta _{\textsf{a}^{r_0 + t} w}< \xi < \theta _{\textsf{a}^{r_0 + t} v}$. For N large enough $r_0 + t$ is at most 2. Therefore both angles are at least $\frac{2}{e^4} N^{-\gamma }$ and at most $\frac{\pi }{2} - \frac{2}{e^4} N^{-\gamma }$. This implies

$$\begin{aligned} \left| \cos (2 \theta _{\textsf{a}^{r_0 + t} v}) - \cos (2 \theta _{\textsf{a}^{r_0 + t} w}) \right| \ge \frac{2}{e^4} N^{-\gamma } \cdot 2(\theta _{\textsf{a}^{r_0 + t} v} - \theta _{\textsf{a}^{r_0 + t} w}) \ge \frac{2}{e^4} N^{-\gamma } \cdot \frac{2c_1^\alpha }{e^4} N^{-\frac{\alpha }{2}} \end{aligned}$$

after an application of Lemma 3.17, S3, Lemma 3.9, and the fac that $r_0 +t \le 2$ for N large enough.

In combination, for all $0 \le t \le N^{-\varpi }$ we get

$$\begin{aligned} \Big | |\!|\textsf{a}^{r_0 + t} v |\!|\cos (2 \theta _{\textsf{a}^{r_0 + t} v}) - |\!|\textsf{a}^{r_0 + t} w |\!|\cos (2 \theta _{\textsf{a}^{r_0 + t} v}) \Big | \le \frac{8}{c_1} N^{0.5 - \delta } \end{aligned}$$

and by S1 together with dilation control

$$\begin{aligned} |\!|\textsf{a}^{r_0 + t} w |\!|\Big | \cos (2 \theta _{\textsf{a}^{r_0 + t} v}) - \cos (2 \theta _{\textsf{a}^{r_0 + t} w}) \Big | \ge \frac{1}{2 e^3 c_2} \frac{N^{0.5}}{\log N} \cdot \frac{4c_1^\alpha }{e^8} N^{-\gamma - \frac{\alpha }{2}} \end{aligned}$$

so that (B.2) gives

$$\begin{aligned} |f_{v,w}'(r_0 + t)| \gg N^{0.5 - \gamma - \alpha } \end{aligned}$$

for all $0 \le t \le N^{-\varpi }$ provided

$$\begin{aligned} \gamma + \alpha < \delta \end{aligned}$$

(R)

holds. But then the mean value theorem implies

$$\begin{aligned} |f_{v,w}(r_0 + N^{-\varpi }) - f_{v,w}(r_0)| \gg N^{0.5 - \gamma - \alpha - \varpi } \end{aligned}$$

which, if one has

$$\begin{aligned} 0.5 - \gamma - \alpha - \varpi > 0.5 - \delta \end{aligned}$$

(R)

implies $|f_{v,w}(r_0 + N^{-\varpi })| > \frac{8}{c_1} N^{0.5 - \delta }$ for N large enough, giving the desired contradiction. $\square$

Since $f_{v,w}$ is strictly decreasing the containment

$$\begin{aligned} \{ 0 \le t \le 1: E_N(v;t) E_N(w;t) = 1 \}\subset & {} \left\{ 0 \le t \le 1: |f_{v,w}(t)| \le \frac{8}{c_1} N^{0.5 - \delta } \right\} \\\subset & {} [r_0, r_0 + N^{-\varpi }] \end{aligned}$$

allows us to conclude that

$$\begin{aligned} \sum _{k=1}^{4\kappa } \sum _{v \in W(k)} \sum _{\begin{array}{c} w \in W \\ w \notin V(k) \end{array}} \int \limits _0^1 E_N(v;t) E_N(w;t) \,\textrm{d}t \ll N^{2-\varpi } \end{aligned}$$

(3.22)

by trivially bounding the sums using (2.1).

3.3 Proof of Theorem 3.1

If—as they may be—the parameters $\alpha , \gamma , \delta , \epsilon , \epsilon ', \zeta , \varpi$ are chosen such that all of the requirements (R) above are satisfied then there is $N_1$ so large that (3.7), (3.14), (3.15), (3.22) together give

$$\begin{aligned} \textsf{m}((3.4)) \le \frac{1}{N^{2(1-\zeta )}} \int \limits _0^1 \left| \sum _{v \in \Lambda (\omega )} E_N(v;t) \right| ^2 \,\textrm{d}\textsf{m}(t) \ll \frac{1}{N^{2(1-\zeta )}} \left( N^{2 - \frac{\alpha }{4}} + N^{2-\gamma } + N^{2-\varpi } \right) \end{aligned}$$

and all $0 \le \phi < 2 \pi$. for all $N \ge N_1$. If

$$\begin{aligned} \zeta < \min \left\{ \tfrac{1}{8} \alpha , \tfrac{1}{2} \gamma , \tfrac{1}{2} \varpi \right\} \end{aligned}$$

(R)

is satisfied then we can find $\lambda > 0$ satisfying the hypothesis.

4 Controlling pairs

In this section we wish to establish (2.10) for all $0 \le \phi < 2 \pi$. Throughout this section, fix $0 \le \phi < 2\pi$ and write $\omega = \textsf{r}^\phi \omega _0$. For the quantity

$$\begin{aligned} \int \limits _0^1 \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \frac{1}{N^2} \sum _{v,w \in \Xi (\textsf{a}^t \omega ;N)} \chi _p(|\!|\textsf{a}^s \textsf{r}^\theta \textsf{h}(v) |\!|) \, \overline{\chi _p(|\!|\textsf{a}^s \textsf{r}^\theta \textsf{h}(w) |\!|)} \,\textrm{d}s \,\textrm{d}\theta \,\textrm{d}t \end{aligned}$$

(4.1)

we wish to produce $\eta > 0$ such that

$$\begin{aligned} (4.1) \ll \dfrac{1}{N^\eta } \end{aligned}$$

(4.2)

for all N large enough.

Writing

$$\begin{aligned} \alpha (u) = \frac{u_1^2 - u_2^2}{|\!|u |\!|} \quad \beta (u) = \frac{2u_1u_2}{|\!|u |\!|} \end{aligned}$$

for any non-zero $u \in \mathbb {R}^2$, we begin by applying the following linearization.

Lemma 4.3

Whenever $|\!|u |\!|\le N^{0.5}$ we have

$$\begin{aligned} \Big | |\!|u |\!|+ \alpha (u) s - |\!|\textsf{a}^s u |\!|\Big | \ll N^{0.5 - 2 \delta } \end{aligned}$$

for all $0 \le s \le S$.

Proof

Using (B.1) the bound

$$\begin{aligned} \left| \frac{f_u''(s)}{2} s^2 \right| < 42 s^2 |\!|u |\!|\ll N^{0.5 - 2 \delta } \end{aligned}$$

gives the desired result via the Lagrange form of the remainder in Taylor’s theorem. $\square$

The requirement

$$\begin{aligned} \delta > 0.25 \end{aligned}$$

(R)

is needed for Lemma 4.3 to be useful. When $\delta > 0.25$ it suffices for (4.2) to produce $\eta > 0$ such that the quantity

$$\begin{aligned}{} & {} \int \limits _0^1 \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \frac{1}{N^2}\nonumber \\{} & {} \quad \sum _{v,w \in \Xi (\textsf{a}^t \omega ;N)} \chi _p(|\!|\textsf{r}^\theta \textsf{h}(v) |\!|+ s \alpha (\textsf{r}^\theta \textsf{h}(v)) ) \, \overline{\chi _p(|\!|\textsf{r}^\theta \textsf{h}(w) |\!|+ s \alpha (\textsf{r}^\theta \textsf{h}(w)))} \,\textrm{d}s \,\textrm{d}\theta \,\textrm{d}t \end{aligned}$$

(4.4)

satisfies

$$\begin{aligned} (4.4) \ll \dfrac{1}{N^\eta } \end{aligned}$$

(4.5)

for all N large enough.

The reduction to (4.5) follows from Lipshitz continuity of $\chi _p$. Indeed, that gives

$$\begin{aligned} | \chi _p(|\!|\textsf{r}^\theta \textsf{h}(v) |\!|+ s \alpha (\textsf{r}^\theta \textsf{h}(v)) ) - \chi _p( |\!|\textsf{a}^s \textsf{r}^\theta \textsf{h}(v) |\!|) | \ll \dfrac{1}{N^{2 \delta - 0.5}} \end{aligned}$$

for all N large enough, whence

$$\begin{aligned} |(4.1) - (4.4)| \ll \dfrac{1}{N^{2 \delta - 0.5}} \end{aligned}$$

(4.6)

for all N large enough.

We now work towards (4.5). Fixing $0 \le t \le 1$ and fixing a parameter $\nu > 0$ to be chosen later ($\nu = \frac{1}{100}$ suffices) we may restrict the summation to those saddle connections v, w for which $|\!|\textsf{h}(v) |\!|,|\!|\textsf{h}(w) |\!|\ge N^{0.5 - \nu }$ both hold.

We may also discard those w for which the angle between v and w is at most $N^{-\frac{\alpha }{2}}$ by an application of Theorem 1.2 similar to the one in Sect. 5; if $C''(\omega ,\epsilon '')$ is the attendant constant ($\epsilon '' = \frac{1}{100}$ suffices) then the requirement

$$\begin{aligned} \frac{1}{2} > \frac{\alpha }{2}(\epsilon '' + 2) \end{aligned}$$

(R)

will allow us to discard as desired.

Fixing v, w satisfying both of these criterion, the quantity

$$\begin{aligned} \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \chi _p(|\!|\textsf{r}^\theta \textsf{h}(v) |\!|+ s \alpha (\textsf{r}^\theta \textsf{h}(v)) ) \, \overline{\chi _p(|\!|\textsf{r}^\theta \textsf{h}(w) |\!|+ s \alpha (\textsf{r}^\theta \textsf{h}(w)))} \,\textrm{d}s \,\textrm{d}\theta \end{aligned}$$

in absolute value is equal to

$$\begin{aligned} \left| \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \chi _p(s \alpha (\textsf{r}^\theta \textsf{h}(v)) ) \, \overline{\chi _p(s \alpha (\textsf{r}^\theta \textsf{h}(w)))} \,\textrm{d}s \,\textrm{d}\theta \right| \end{aligned}$$

because rotations do not change the lengths of holonomy vectors.

Define $A(v,w) \ge 0$ by $A(v,w)^2 = (\alpha (\textsf{h}(v)) - |\!|\textsf{h}(w) |\!|)^2 + \beta (\textsf{h}(v))^2$. To proceed we quote the following estimate from [1].

Lemma 4.7

We have

$$\begin{aligned} \left| \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \chi _p(s \alpha (\textsf{r}^\theta \textsf{h}(v)) ) \, \overline{\chi _p(s \alpha (\textsf{r}^\theta \textsf{h}(w)))} \,\textrm{d}s \,\textrm{d}\theta \right| \le \frac{4}{\pi N^{0.5}} + \frac{N^\delta }{\pi A(v,w)} \left( \log N + \log \frac{\pi }{4} \right) \end{aligned}$$

for all N large enough.

Proof

The estimate follows by duplicating (with $R = N^{0.5}$ and $\epsilon = 1$) the proof of [1, Lemma 12] up to [1, Equation (24)] and the estimates immediately after [1, Equation (24)]. That much of the proof does not use the hypothesis. $\square$

Our assumptions on $\Vert \textsf{h}(v) \Vert$, $\Vert \textsf{h}(w) \Vert$ and the angle between v and w give

$$\begin{aligned} A(v,w) \ge |\!|\textsf{h}(v) |\!|| \sin (2(\theta _v - \theta _w)) | \ge N^{0.5 - \nu } \cdot c_1^\alpha N^{-\frac{\alpha }{2}} \end{aligned}$$

so that overall

$$\begin{aligned} \left| \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \chi _p(s \alpha (\textsf{r}^\theta \textsf{h}(v)) ) \, \overline{\chi _p(s \alpha (\textsf{r}^\theta \textsf{h}(w)))} \,\textrm{d}s \,\textrm{d}\theta \right| \ll N^{-0.5} + N^{\delta + \nu + \frac{\alpha }{2} - 0.5} \log N \end{aligned}$$

for our vectors v, w. This is satisfacory provided

$$\begin{aligned} \delta + \nu + \frac{\alpha }{2} < 0.5 \end{aligned}$$

(R)

holds, as we may then take $\eta = \delta + \nu + \frac{\alpha }{2} - 0.5$ to establish (4.5).

5 Proof of main theorem

In Sect. 2 we reduced (via Theorem 2.8) the proof of Theorem 1.1 to the statement that

$$\begin{aligned} \dfrac{1}{2\pi } \int \limits _0^{2\pi } \int \limits _0^1 \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \frac{1}{N^2} \sum _{v,w \in \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \textsf{r}^\phi \omega _0;N)} \chi _p(|\!|\textsf{h}(v) |\!|) \, \overline{\chi _p(|\!|\textsf{h}(w) |\!|)} \,\textrm{d}s \,\textrm{d}\theta \,\textrm{d}t \ll \frac{1}{N^\eta } \end{aligned}$$

(5.1)

for some $\eta > 0$. We finish here the proof of Theorem 1.1 by establishing (5.1).

Lemma 5.2

In order to prove (5.1) it suffices to prove

$$\begin{aligned} \int \limits _0^1 \frac{1}{2\pi } \int \limits _0^{2\pi } \frac{1}{S} \int \limits _0^{S} \frac{1}{N^2} \sum _{v,w \in \Xi (\textsf{a}^t \omega ;N)} \chi _p(|\!|\textsf{h}(\textsf{a}^s \textsf{r}^\theta v) |\!|) \, \overline{\chi _p(|\!|\textsf{h}(\textsf{a}^s \textsf{r}^\theta w) |\!|)} \,\textrm{d}s \,\textrm{d}\theta \,\textrm{d}t \ll \frac{1}{N^\eta } \end{aligned}$$

(5.3)

for some $\eta > 0$.

Proof

Let $\Omega$ be the set (3.4). The conclusion of Theorem 3.1 is that $\textsf{m}(\Omega ) \ll N^{-\lambda }$. When t does not belong to $\Omega$ we have

$$\begin{aligned} \left| \Xi (\textsf{a}^s \textsf{r}^\theta \textsf{a}^t \omega ; N) \triangle \textsf{a}^s \Xi (\textsf{r}^\theta \textsf{a}^t \omega ;N) \right| \le N^{1 - \zeta } \end{aligned}$$

for all $0 \le s \le S$ and all $0 \le \theta < 2\pi$. It follows that $N^{-\zeta } + N^{-\lambda }$ controls the difference between (5.1) and (5.3). $\square$

To apply the material of Sect. 3—which establishes Theorem 3.1—and the material of Sect. 4—which establishes (5.3)—we need to ensure that the requirements (R) above can be satisfied simultaneously. The choices

$$\begin{aligned} \delta = \frac{1}{3} \qquad \alpha = \epsilon = \epsilon ' = \epsilon '' = \nu = \varpi = \frac{1}{100} \qquad \gamma = \frac{1}{300} \qquad \zeta = \frac{1}{1000} \end{aligned}$$

show that this is possible, concluding the proof of Theorem 1.1.

Appendices

Appendix A: Proof of Theorem 1.2 on counting in sectors by Benjamin Dozier

Theorem 1.2 is a more explicit version of [2, Theorem 1.8]. Getting the explicit version is a matter of keeping careful track of the constants in various proofs in that paper. The first step is to give explicit constants in [2, Proposition 2.1, p. 94].

Proposition A.1

Fix ${\mathcal {H}}$ and $0< \delta < 1/2$. Define $\alpha : {\mathcal {H}} \rightarrow \mathbb {R}$ by $\alpha (\omega ) = 1/\ell (\omega )^{1+\delta }$. There is a constant b such that for any interval $I \subset \textsf{S}^1$ there is a constant $c_I = O\left( \frac{1}{|I|^{1-2\delta }}\right)$ such that for any $\omega \in {\mathcal {H}}$

$$\begin{aligned} \int \limits _I \dfrac{1}{\ell (\textsf{a}^t \textsf{r}^\theta \omega )^{1+\delta }} \,\textrm{d}\theta < c_I \cdot e^{-(1-2\delta )T} \alpha (\omega ) + b \cdot |I| \end{aligned}$$

for all $T \ge 0$.

The proof of Proposition A.1 is at the end of the appendix. We first prove Theorem 1.2 assuming Proposition A.1.

Proof of Theorem 1.2 assuming Proposition A.1

From Proposition A.1, we get that $c_{I'}=c_I = O \left( \frac{1}{|I|^{1-2\delta }}\right)$, where the implied constant depends on only on genus of the surface. We will also use $\alpha (X) = 1/\ell (X)^{1+\delta }$. For our lower bound on R, we can then take

$$\begin{aligned} R_0=O \left( \frac{1}{|I|^{(2-2\delta )/(1-2\delta )}} \cdot \frac{1}{\ell (X)^{(1+\delta )/(1-2\delta )}} \right) . \end{aligned}$$

Since we can choose $\delta$ as small as we wish (in particular, we can take $\delta = \epsilon /(2+2\epsilon )$), we get for every $\epsilon >0$,

$$\begin{aligned} R_0= \frac{C(X,\epsilon )}{|I|^{2+\epsilon }}, \end{aligned}$$

where

$$\begin{aligned} C(X,\epsilon ) = O\left( \frac{1}{\ell (X)^{(1+(\epsilon /(2+2\epsilon ))/(1-2 \epsilon /(2+2\epsilon ))}}\right) . \end{aligned}$$

We claim that we can take $C(X,\epsilon )$ to depend continuously on X. It suffices to show that $\ell (X)$, the length of the shortest saddle connection, depends continuously on X. We claim that $\ell$ equals the “flat systole function” f, which is defined as the length of the shortest curve or arc (starting/ending at zeros) that is not homotopic to a point (relative to zeros, in the case of an arc). Clearly $f\le \ell$, since any saddle connection is such an arc. To see that $\ell \le f$, note that any such curve/arc can be tightened to either (i) a union of saddle connections, or (ii) a closed geodesic, parallel copies of which form a cylinder. In case (i), picking any one of the saddle connections in the union gives a saddle connection that has length at most that of the original curve/arc. In case (ii), there is a saddle connection on the boundary of the cylinder that has length at most that of the original curve/arc. Finally, the flat systole function clearly varies continuously, hence so does $\ell$. $\square$

Proof of Proposition A.1

The result follows from the following modifications to [2].

• Explicit constants in [2, Proposition 5.5, p.111]: We can take $c_I = c_I^{(k)} = O_k\left( \frac{1}{|I|^{1-2\delta }}\right)$.

• Addendum to [2, Proof of Proposition 2.1 assuming Proposition 5.5, p. 111]: Note that by the definition of $\alpha _i$, we have $\alpha _1(X) \ge \alpha _i(X)$, for each i and every X (this is because $\alpha _i(X)$ is defined in terms of the saddle connections on the boundary of a complex of complexity i; any such saddle connection forms a complex of complexity 1, and thus is included in the definition of $\alpha _1$.) Thus we can replace the sum $\sum _{j\ge k}\alpha _j(X)$ that we get from [2, Proposition 5.5] with $M \cdot \alpha _1(X) = M/\ell (X)^{1+\delta }$, where M is the complexity of X (and then we can absorb the constant M into the constant $c_I)$.

• Explicit constants in [2, Lemma 5.8, p. 114]: We can take $t_0 \approx \tau + \log \frac{1}{|I|}$.

• Addendum to [2, Proof of Proposition 5.5, pp. 116–118]:

  • We can take $m \approx t_0(\tau ,|I|)/\tau \approx \frac{1}{\tau } \log \frac{1}{|I|}$.

  • We can take $w_{\tau ,I}^{(k)} = c_{I}^{(k+1)}c_2w_{\tau }$.

  • We can take $c_{\tau ,m} =\left( e^{-\tau (1-2\delta )}\right) ^{-m+1} + w_{\tau ,I} \cdot e^{\tau (1-2\delta )}.$ From above, we have $w_{\tau ,I}^{(k)} = c_{I}^{(k+1)}c_2w_{\tau }$. For the maximal possible k, there are no terms from higher complexity, i.e. all the higher $\alpha _i$ are 0, so, for this k, we get

    $$\begin{aligned} c_{\tau ,I}^{(k)} = O_{k}\left( \left( e^{-\tau (1-2\delta )}\right) ^{-m+1} \right) =O_k \left( e^{\tau (1-2\delta )\frac{1}{\tau }\log \frac{1}{|I|}} \right) = O_k \left( \frac{1}{|I|^{1-2\delta }} \right) . \end{aligned}$$

    Then inducting down by complexity, we get the same result for all k (with different implied constant in the $O_k$).

$\square$

Appendix B: Length function

In this appendix we collect various simple results about the function $f_v(t) = |\!|\textsf{a}^t v |\!|$ defined on $\mathbb {R}$ for any vector $v \in \mathbb {R}^2$ with positive entries, and its relative $f_{v,w} = f_v - f_w$. We have

$$\begin{aligned} f_v(t) = \sqrt{e^{2t} v_1^2 + e^{-2t} v_2^2} = f_{\textsf{a}^t v}(0) \end{aligned}$$

and define

$$\begin{aligned} h_v(t) = e^{2t} v_1^2 - e^{-2t} v_2^2 = h_{\textsf{a}^t v}(0) \end{aligned}$$

for convenience. First note that

$$\begin{aligned} f_v'(t) = \frac{h_v(t)}{f_v(t)} \qquad h_v'(t) = 2 f_v(t)^2 \end{aligned}$$

from which

$$\begin{aligned} f_v''(t) = \frac{2f_v(t)^4 - h_v(t)^2}{f_v(t)^3} = f_v(t) + \frac{4 v_1^2 v_2^2}{f_v(t)^3} \end{aligned}$$

(B.1)

follows.

Writing $V = \textsf{a}^t v$ and $\theta _V$ for its argument we can write

$$\begin{aligned} f_v'(t) = \frac{V_1^2 - V_2^2}{|\!|V |\!|} = |\!|V |\!|\Big ( (\cos \theta _V)^2 - (\sin \theta _V)^2 \Big ) = |\!|V |\!|\cos 2 \theta _V \end{aligned}$$

(B.2)

and

$$\begin{aligned} f_v''(t)= & {} \frac{2 (V_1^2 + V_2^2)^2 - (V_1^2 - V_2^2)^2}{|\!|V |\!|^3} = \frac{|\!|V |\!|^4 + 4 V_1^2 V_2^2}{|\!|V |\!|^3} = |\!|V |\!|+ \frac{4V_1^2 V_2^2}{|\!|V |\!|^3} \\ {}= & {} |\!|V |\!|\Big (1 + (\sin 2 \theta _V)^2 \Big ) \end{aligned}$$

for all $t \in \mathbb {R}$ from $2 V_1 V_2 = |\!|V |\!|^2 \sin 2 \theta _V$. These calculations show $f_v$ is concave everywhere with a global minimum at

$$\begin{aligned} m(v) = \frac{1}{2} \log \frac{v_2}{v_1} \end{aligned}$$

where $\theta _V = \frac{\pi }{4}$.

We now turn to $f_{v,w}$ assuming $v \ne w$. We assume without loss of generality that v and w are in the first quadrant. If $f_{v,w}$ has a zero it must be at

$$\begin{aligned} r(v,w) = \frac{1}{4} \log \frac{v_2^2 - w_2^2}{w_1^2 - v_1^2} \end{aligned}$$

(B.3)

giving the following fac.

Fact B.4

• If $w_1 > v_1$ and $v_2 > w_2$ the function $f_{v,w}$ has a unique zero.

• If $v_1 > w_1$ and $w_2 > v_2$ the function $f_{v,w}$ has a unique zero.

• In no other case does $f_{v,w}$ have a zero.

Fact B.5

If $f_{v,w}$ has a zero it is either strictly increasing or strictly decreasing. In other words, if $f_{v,w}$ has a zero then $f_{v,w}'$ does not.

Proof

Switching the roles of v and w if necessary, we many assume $w_1 > v_1$ and $v_2 > w_2$. We then have

$$\begin{aligned} m(w) = \frac{1}{2} \log \frac{w_2}{w_1} < \frac{1}{2} \log \frac{v_2}{v_1} = m(v) \end{aligned}$$

and on the interval (m(w), m(v)] the function $f_w$ is strictly increasing while on [m(w), m(v)) the function $f_v$ is strictly decreasing. Thus $f_{v,w}$ is strictly decreasing on [m(w), m(v)].

Next, note that since $v_2 > w_2$ we have $f_{v,w}(t) > 0$ for $t < r(v,w)$ and $f_{v,w}(t) < 0$ for $t > r(v,w)$.

Consider the case that $m(w) \le r(v,w)$. Thus we have $|\!|W |\!|\le |\!|V |\!|$ and $\theta _V > \theta _W \ge \frac{\pi }{4}$ for all $t \le m(w)$. Accordingly

$$\begin{aligned} f_v'(t) = |\!|V |\!|\cos (2 \theta _V) < |\!|W |\!|\cos (2 \theta _W) = f_w'(t) \end{aligned}$$

and $f_{v,w}'$ is negative on $(-\infty ,m(w)]$ whence $f_{v,w}$ is strictly decreasing on $(-\infty , m(v)]$.

In the case that $r(v,w) < m(w)$ then the above argument shows only that $f_{v,w}'$ is negative on $(-\infty ,r(v,w)]$. On the interval (r(v, w), m(w)) we have $|\!|W |\!|> |\!|V |\!|$ and $\theta _V> \theta _W > \frac{\pi }{4}$ giving $f_v'' < f_w''$ thereon. Thus we may extend negativity of $f_{v,w}'$ to $(-\infty ,m(w)]$ and $f_{v,w}$ is again strictly decreasing on all of $(-\infty ,m(v)]$.

The cases $r(v,w) \le m(v)$ and $m(v) < r(v,w)$ are similar to the above, and altogether $f_{v,w}$ is strictly decreasing on all of $\mathbb {R}$. $\square$