Transcendence criteria for multidimensional continued fractions

TL;DR

This paper develops new transcendence criteria for multidimensional continued fractions, extending classical Diophantine approximation results to higher dimensions and identifying conditions under which these fractions are transcendental.

Contribution

It introduces novel transcendence criteria for multidimensional continued fractions, including Liouville-type and quasi-periodic cases, and provides bounds on heights of cubic irrationals.

Findings

Liouville-type multidimensional continued fractions are transcendental.

Quasi-periodic multidimensional continued fractions are transcendental.

Upper bounds on the height of cubic irrationals from periodic continued fractions.

Abstract

Classical results on Diophantine approximation, such as Roth's theorem, provide the most effective techniques for proving the transcendence of special kinds of continued fractions. Multidimensional continued fractions are a generalization of classical continued fractions, introduced by Jacobi, and there are many well-studied open problems related to them. In this paper, we establish transcendence criteria for multidimensional continued fractions. In particular, we show that some Liouville-type and quasi-periodic multidimensional continued fractions are transcendental. We also obtain an upper bound on the naive height of cubic irrationals arising from periodic multidimensional continued fractions and exploit it to prove the transcendence criteria in the quasi-periodic case.