A geometric proof of Lagrange's theorem for continued fractions

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 $i+1/i=0$.

💡 Research Summary

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The paper presents a unified geometric framework that extends the classical results on regular continued fractions—namely, “finite expansions correspond to rational numbers” and “eventually periodic expansions correspond to quadratic surds”—to a broad family of continued‑fraction algorithms called proper and discrete Iwasawa continued fractions. These algorithms encompass nearest‑integer real continued fractions, Hurwitz complex continued fractions, as well as higher‑dimensional and non‑commutative cases such as 3‑dimensional, quaternionic, octonionic, and Heisenberg continued fractions.

The authors first define an Iwasawa continued‑fraction system by four ingredients: (i) an ambient space (X) built from a real associative division algebra (k) (ℝ, ℂ, ℍ, or 𝕆) and a dimension (n); (ii) a lattice (Z) of isometries of (X) that provides the “digits”; (iii) a fundamental domain (K\subset X) whose closure lies inside the open unit ball (the “proper” condition); and (iv) a Korányi inversion (\iota) that expands uniformly on (K). When the pair ((X,Z,\iota,K)) satisfies properness and discreteness, the modular group (\mathcal M=\langle Z,\iota\rangle) is a discrete subgroup of the isometry group of the associated rank‑one symmetric space (a hyperbolic space modeled by Iwasawa coordinates).

The forward‑shift map (T:K\to K) is defined by (T(x)=\iota x-