Addendum to "On triangular Billard"

TL;DR

This paper classifies all integer triples where no integer a satisfies a specific modular inequality involving n, s, and t, thereby completing previous research and closing a gap in the mathematical understanding of triangular billiards.

Contribution

It provides a complete classification of cases with no solutions to a particular modular inequality, extending and completing earlier work in the field.

Findings

Identifies all integer triples (n, s, t) with the specified property.

Completes the classification previously incomplete in the literature.

Closes a gap in the mathematical theory of triangular billiards.

Abstract

Let $n, s, t$ be integers satisfying $(n,s,t)=1$. We classify all cases such that there is no integer $a$ with $n/2<as\bmod n+at\bmod n<3n/2$. This closes a gap in previous work of the author (Comment Math. Helv. 76, 501--505).