Uniform distribution of saddle connection lengths

By a flat surface we mean a pair $(X,\omega)$ where $X$ is a closed, compact Riemann surface of genus $g\geq 2$ and $\omega$ is a non-zero holomorphic one form on $X$. Every non-zero holomorphic one form on $X$ has $2g-2$ zeros where $g$ is the genus of $X$. Fixing the genus of $X$ the set $\{(X,\omega):\omega\neq 0\}$ is partitioned into strata according to the possible partitions of $2g-2$. Each connected component of a stratum carries a natural volume measure that is invariant under the action of $\mathsf{SL}(2,\mathbb{R})$ on pairs $(X,\omega)$. The $\mathsf{SL}(2,\mathbb{R})$ action preserves the area of $(X,\omega)$. Masur [Mas82] and Veech [Vee82] showed independently that the natural volume on a connected component induces on its subset $\mathcal{H}$ of unit area flat surfaces a probability measure $\mathsf{M}$ that is ergodic for the $\mathsf{SL}(2,\mathbb{R})$ actions.

Fix a flat surface $(X,\omega)$. Every non-zero holomorphic one form gives, away from its finite set $\Sigma$ of zeroes, an atlas of charts to $\mathbb{R}^{2}$ whose transition maps are translations in $\mathbb{R}^{2}$. For every $0\leq\theta<\pi$ we can use such an atlas to induce a foliation on $X\setminus\Sigma$ given locally by the lines in $\mathbb{R}^{2}$ making an angle of $\theta$ with the horiontal axis. We can also use such an atlas to measure lengths of curves on $X$. In particular we can calculate the lengths of the leaves in the above foliations. A saddle connection of $(X,\omega)$ is any leaf of any of these foliations that starts and ends at a point of $\Sigma$. The holonomy of a saddle connection $v$ is the vector

in $\mathbb{R}^{2}$.

In this paper we are interested in the uniform distribution of saddle connection lengths. Recall that a sequence $n\mapsto x_{n}$ in $\mathbb{R}$ is uniformly distributed mod 1 when

for all intervals $[a,b)\subset[0,1)$. The well-known Weyl criterion is that uniform distribution of $n\mapsto x_{n}$ is equivalent to

for all $p\in\mathbb{Z}\setminus\{0\}$ and one can use it to prove, for example, that $n\mapsto n\alpha$ is uniformly distributed for all irrational $\alpha$.

Returning to flat surfaces (and suppressing $X$ from our notation) we write $\Lambda(\omega)$ for the set of saddle connections of $\omega$. For each $R>0$ write $\Lambda(\omega;R)$ for the set of saddle connections whose holonomy vectors have length at most $R$. Given also $S>0$ write $\Lambda(\omega;R,S)$ for the set of saddle connections whose holonomy vectors lie within the ellipse $(Ry)^{2}+(Sx)^{2}=(RS)^{2}$.

It follows from minor modifications to work of Vorobets [Vor05, Theorem 1.9] that the projection of $\Lambda(\omega;R)$ to the unit circle along rays through the origin is uniformly distributed as $R\to\infty$ for almost every $\omega$. Specifically, by [Vor05, Proposition 4.5] it suffices to show that the set of saddle connection holonomies satisfy axioms (0), (A), (B), (C) and (E) therein. That saddle connections doe satisfy these axioms follows from the definitions and results in [EM01] just as it does for the sets Vorobets considers.

Our main result is that lengths of saddle connection holonomy vectors are also almost always uniformly distributed mod 1. To make sense of this we fix, for each flat surface $\omega$, an enumeration $n\mapsto v_{n}$ of $\Lambda(\omega)$ such that $n\mapsto|\!|_{\mathsf{}}\mathsf{h}(v_{n})|\!|_{\mathsf{}}$ is non-decreasing.

Masur [Mas88, Mas90] proved that for every surface $\omega$ there exists $c_{1}$, $c_{2}$ (depending on $\omega$) so that for all large $R$ we have

(see also Vorobets [Vor03]). Veech [Vee98] introduced what is now called the Siegel-Veech transform to show that there exists $c$ depending only on $\nu$ and $R_{n}\to\infty$ so that

for $\mathsf{M}$ almost every $\omega$. He also showed the limit exists for all surfaces in certain suborbifolds of $\mathcal{H}$. Building on Veech’s approach, Eskin-Masur [EM01] proved that for every $\mathsf{SL}(2,\mathbb{R})$ invariant and $\mathsf{SL}(2,\mathbb{R})$ ergodic probability measure $\mathsf{m}$ there exists $c$ so that we have

for $\mathsf{m}$ almost every $\omega$. Dozier [Doz17a] proved analogues of these results for saddle connections with holonomies in certain sectors of directions.

The best known results for asymptotic counting that hold on all surfaces are Eskin, Mirzakhani and Mohammadi’s result [EMM15, Theorem 2.12] that every surface has an asymptotic growth of saddle connections on average, and Dozier’s result [Doz17] building on techniques in [EM11] that the constants $c_{1},c_{2}$ in (2) can be chosen to depend only on the connected component of the stratum in question. The best almost everywhere counting result is the following theorem of Nevo, Rühr and Weiss.

The gaps of saddle connection directions has also been studied by Athreya, the first named author, Lelievre, Uyanik and Work [AC12], [ACL15] and [UW16]. Recently Athreya, Cheung and Masur [ACM17] have begun investigating the $\mathrm{L}^{2}$ properties of the Siegel-Veech transform.

Most of these papers to a greater or lesser extent adhere to the philosophy of renormalization dynamics, using the $\mathsf{SL}(2,\mathbb{R})$ action and especially the geodesic flow

to translate questions about long saddle connections to moderate saddle connections on different surfaces. Our approach is a little different: we still use the $\mathsf{SL}(2,\mathbb{R})$ action, but only as a means to perturb. Namely, we do not apply the geodesic flow $\mathsf{g}^{t}$ long enough to make a saddle connection $\mathsf{O}(1)$ in length. Instead, writing also

we prove that for almost every $\omega$ and for almost every $\theta,t$ the lengths of saddle connections on $\mathsf{g}^{t}\mathsf{r}^{\theta}\omega$ are uniformly distributed mod 1. However, we emphasize that we do use Theorem 4, which does use renormalization dynamics.

We remark that even if the conclusion of [NRW17, Theorem 1.1] were known for every flat surface, our methods would not give uniform distribution of saddle connection lengths for every flat surface. Therefore, the following question is in general open.

However, in the presence a very strong error term for the saddle connection counting function $|\Lambda(\omega;R)|$ one can obtain uniform distribution of saddle connection lengths everywhere with a much simpler proof. The appendix by Daniel El-Baz and Bingrong Huang proves using Huxley and Nowak’s bound [HN96] on the counting function for primitive lattice points that the conclusion of Theorem 1 holds for every torus.

The research of J. Chaika was supported in part by NSF grants DMS-135500 and DMS-1452762, the Sloan foundation, a Poincarè chair, and a Warnock chair. J. Chaika thanks Alex Eskin for encouraging him to pursue this question. The research of D. Robertson was supported by NSF grant DMS-1703597.

In this section we prove Theorem 1. Let $\mathcal{H}$ be the set of unit-area flat surfaces in the connected component of a stratum and write $\mathsf{m}$ for the Masur-Veech probability measure $\mathsf{m}$ on $\mathcal{H}$. Let $c>0$ and $\kappa>0$ be as in Theorem 4. Our proof of Theorem 1 has the following four ingredients.

1. 1.

   The Weyl criterion.

2. 2.

   An exhaustion by compact sets.

3. 3.

   A relation with orbit averages.

4. 4.

   A linear approximation to handle exponential sums.

These steps are covered in the following four subsections. In Subsection 2.5 we combine the steps to prove Theorem 1.