# Density of convex billiards with rational caustics

## TL;DR

This paper proves that within the space of convex billiard boundaries, those with rational caustics are densely distributed, with density depending on the smoothness of the boundary.

## Contribution

It establishes the density of convex billiard boundaries with rational caustics, showing polynomial density in smooth cases and exponential density in analytic cases.

## Key findings

- Boundaries with rational caustics are dense in convex billiards.
- Density is polynomial in smooth cases.
- Density is exponential in analytic cases.

## Abstract

We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number $1/q$ is polynomial sense in the smooth case, and exponentially dense in the analytic case.