Linear recurrence sequences and periodicity of multidimensional continued fractions

TL;DR

This paper characterizes the periodicity of multidimensional continued fractions, specifically the Jacobi--Perron algorithm, using linear recurrence sequences, thus generalizing classical continued fraction results.

Contribution

It provides a novel characterization linking periodicity of multidimensional continued fractions to linear recurrence sequences of convergents.

Findings

Partial quotients are periodic iff numerators and denominators are linear recurrence sequences.

Generalizes classical continued fraction periodicity results to multidimensional cases.

Offers a theoretical framework for understanding periodicity in multidimensional continued fractions.

Abstract

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. However, there does not exist any algorithm that provides a periodic multidimensional continued fraction when algebraic irrationalities are given as inputs. In this paper, we provide a characterization for periodicity of Jacobi--Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerators and denominators of convergents are linear recurrence sequences, generalizing similar results that hold for classical continued fractions.