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A geometric proof of Lagrange's theorem for continued fractions

Anton Lukyanenko, Joseph Vandehey

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 $i+1/i=0$.

A geometric proof of Lagrange's theorem for continued fractions

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 : has a finite expansion, for the appropriate modular group , and is a fixed point of a parabolic transformation in . Eventually-periodic points correspond exactly to fixed points of loxodromic elements of , 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 .
Paper Structure (12 sections, 19 theorems, 21 equations, 1 figure)

This paper contains 12 sections, 19 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The real case
  3. Organization
  4. Acknowledgements
  5. Geometry of Iwasawa CFs
  6. Proof of the Main Theorem
  7. Loxodromic fixed points
  8. Real and complex cases
  9. Clifford algebra formalism
  10. Loxodromic fixed points in $\mathbb{R}^3$
  11. Loxodromic fixed points in $\mathbb{R}^4$
  12. A note on quaternionic identities

Key Result

Theorem 1.1

Let $(\mathbb X, \mathcal{Z}, \iota, K)$ be a proper and discrete Iwasawa CF. Then $x\in X$ has a finite expansion if and only if it is a fixed point of a parabolic element of $\mathcal{M}=\langle\mathcal{Z}, \iota\rangle$, and has an eventually-periodic expansion if and only if it is a fixed point

Figures (1)

  • Figure 1: Geometric setup for the proof of Lagrange's theorem for nearest-integer CFs, under the simplifying assumption that $(x,x')$ are widely spaced, i.e. $|x|<0.5$ and $|x'|>\sqrt{2}$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • proof : Sketch of the proof of Theorem \ref{['thm:generalmain']} for nearest-integer CFs
  • Remark 1.5
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 35 more