Transcendence criteria for multidimensional continued fractions
Federico Accossato, Nadir Murru, Giuliano Romeo
TL;DR
This work extends transcendence theory to multidimensional continued fractions by developing criteria for Jacobi and Jacobi–Perron MCFs, including Liouville-type and quasi-periodic cases. It combines convergence control, explicit height bounds for cubic irrationals from periodic MCFs, and Diophantine approximation techniques (e.g., Roth-type results) to establish transcendence under precise growth conditions of partial quotients and blocks. A key achievement is the explicit height bound $H(\alpha_0), H(\beta_0) \le 3024\,C_{h+k-1}^9$ for $0<\alpha_0,\beta_0<1$ in the periodic Jacobi setting, along with a general framework for proving transcendence in Liouville-type and quasi-periodic MCFs. The results pave the way for constructing higher-dimensional transcendental numbers via MCFs, while identifying significant open problems, including higher-dimensional height bounds, independence questions, and extensions to automatic-type multidimensional 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.