The Distribution of Gaps for Saddle Connection Directions

Abstract

Abstract. Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential ω on a Riemann surface of genus g ≥ 2 the smallest gap between saddle connection directions of length at most a fixed length decays faster than quadratically in the length. We also characterize the exceptional set: the decay rate is not faster than quadratic if and only if ω is a lattice surface.

Gap distributions and homogeneous dynamics

“…To each saddle connection γ one can associate a holonomy vector v γ = γ ω ∈ C. The set of holonomy vectors Λ sc (ω) is a discrete subset of C ∼ = R 2 , and varies equivariantly under the natural SL(2, R) action on the set of translation surfaces. Motivated by such concerns, and inspired by the work of Marklof-Strombergsson [14] (of which more below in §1.6), the author and J. Chaika [2] studied the gap distribution for saddle connection directions. The relationship between flat surfaces and billiards in polygons is given by a natural unfolding procedure, which associates to each (rational) polygon P a translation surface (X P , ω P ).…”

Spatial Statistics of Apollonian Gaskets

“…Based on these findings, we make the following conjecture. for any s P r0, 8q, where F T,R psq is defined in (1). Moreover, the function F is independent of the chosen Apollonian gasket.…”

“…Spatial statistics from some other point processes have otherwise been rigorously established: gap distribution of the fractional parts of p ? nq by Elkies and McMullen [8], distribution of directions of Euclidean or hyperbolic lattices [6], [5], [13], [21], [17], distribution of Farey sequences [12], [4], [3], and gap distribution of saddle connection directions in translation surfaces [1], [2]. Our list of interesting works here is far from inclusive.…”

The gap distribution of slopes on the golden L

“…Typical surfaces. In [2], the first two authors considered gap distributions for typical translation surfaces, that is, surfaces in a set of full measure for the Masur-Veech measure (or, in fact, any ergodic SL(2, R)-invariant measure) on a connected component of a stratum of the moduli space Ω g of genus g ≥ 2 translation surfaces. It was shown that the limiting distribution exists, and is the same for almost every surface, and, as above, the tail is quadratic.…”

“…This is essentially equivalent to the no small triangles condition of Smillie-Weiss [20]. However, the only explicit computations in [2] were for branched covers of tori, and relied on previous work of Marklof-Strömbergsson [15] on the space of affine lattices. To the best of our knowledge, the current paper gives the first explicit computation of a gap distribution of saddle connections for a surface which is not a branched cover of a torus.…”