A general Lagrange Theorem

TL;DR

This paper presents a simple, broadly applicable proof of the Lagrange Theorem, establishing that a number has an eventually periodic continued fraction expansion if and only if it is a quadratic irrational, within a general dynamical systems framework.

Contribution

It provides a simple, nearly computation-free proof of the general Lagrange Theorem applicable to various continued fraction variants within a dynamical systems context.

Findings

The Lagrange Theorem holds for all variants of continued fractions.

A simple proof can be derived without deep results from interval exchange transformations.

The theorem characterizes quadratic irrationals via periodic expansions.

Abstract

The ordinary continued fractions expansion of a real number is based on the Euclidean division. Variants of the latter yield variants of the former, all encompassed by a more general Dynamical Systems framework. For all these variants the Lagrange Theorem holds: a number has an eventually periodic expansion if and only if it is a quadratic irrational. This fact is surely known for specific expansions, but the only proof for the general case that I could trace in the literature follows as an implicit corollary from much deeper results by Boshernitzan and Carroll on interval exchange transformations. It may then be useful to have at hand a simple and virtually computation-free proof of a general Lagrange Theorem.