Uniform distribution of saddle connection lengths

TL;DR

This paper proves that for almost all flat surfaces, the sequence of saddle connection lengths, when ordered and considered modulo one, is uniformly distributed, revealing a fundamental statistical property of these geometric structures.

Contribution

It establishes the uniform distribution of saddle connection lengths for almost all flat surfaces, a significant result in the study of flat geometry and dynamical systems.

Findings

Saddle connection lengths are uniformly distributed mod one for almost all flat surfaces.

The result applies to a broad class of flat surfaces, indicating a universal statistical behavior.

Provides insights into the geometric and dynamical properties of flat surfaces.

Abstract

For almost every flat surface the sequence of saddle connection lengths listed in increasing order is uniformly distributed mod one.