Lagrange's Theorem for continued fractions on the Heisenberg group

TL;DR

This paper establishes an analog of Lagrange's Theorem for continued fractions within the Heisenberg group, linking periodic expansions to solutions of specific quadratic forms, thus extending classical number theory results to a non-commutative setting.

Contribution

It introduces a new characterization of periodic continued fractions in the Heisenberg group, connecting them to quadratic forms, which is a novel extension of classical theory.

Findings

Points with eventually periodic continued fractions satisfy a quadratic form.

Conversely, solutions to the quadratic form have periodic continued fractions.

The theorem generalizes Lagrange's classical result to a non-commutative setting.

Abstract

We prove an analog of Lagrange's Theorem for continued fractions on the Heisenberg group: points with an eventually periodic continued fraction expansion are those that satisfy a particular type of quadratic form, and vice-versa.