A geometric proof of Lagrange's theorem for continued fractions

TL;DR

This paper extends Lagrange's theorem to various types of continued fractions, including complex and quaternionic, providing a geometric proof and characterizing finite and periodic expansions via fixed points of transformations.

Contribution

It introduces a geometric framework for Lagrange's theorem applicable to diverse continued fraction systems, including higher-dimensional and non-commutative cases.

Findings

Finite expansions correspond to fixed points of parabolic transformations.

Eventually-periodic points are fixed points of loxodromic elements.

Provides a geometric proof of Lagrange's theorem for multiple CF types.

Abstract

For regular continued fractions (CFs), points with finite expansions are exactly the rationals and, by Lagrange's theorem, points with eventually-periodic expansions are exactly the roots of non-degenerate quadratic equations with integer coefficients. We extend both results to proper and discrete Iwasawa CFs, including real, complex, 3D, quaternionic, octonionic, and Heisenberg CFs. Namely, the following three conditions are equivalent for a point $p$: $p$ has a finite expansion, $p\in \mathcal M(\infty)$ for the appropriate modular group $\mathcal M$, and $p$ is a fixed point of a parabolic transformation in $\mathcal M$. Eventually-periodic points correspond exactly to fixed points of loxodromic elements of $\mathcal M$, which can be interpreted as roots of non-degenerate quadratics using the Clifford Algebra formalism of Ahlfors. In particular, this provides a new geometric proof of Lagrange's theorem for nearest-integer real CFs and Hurwitz complex CFs. Lastly, we comment on generalizations of the identity $i+1/i=0$.