On the Local-Global Conjecture for Combinatorial Period Lengths of Closed Billiards on the Regular Pentagon

TL;DR

This paper investigates the distribution of combinatorial lengths of asymmetric periodic billiard trajectories in a regular pentagon, proving a density-one result that advances understanding of the conjecture by Davis-Lelievre.

Contribution

It provides a density-one proof for a conjecture regarding combinatorial lengths of trajectories in regular pentagon billiards, a significant step in the field.

Findings

Proves a density-one version of the Davis-Lelievre conjecture.

Establishes the distribution of combinatorial lengths for asymmetric trajectories.

Advances the understanding of periodic billiard trajectories in regular polygons.

Abstract

We study the set of combinatorial lengths of asymmetric periodic trajectories on the regular pentagon, proving a density-one version of a conjecture of Davis-Lelievre.