Geometry of Continued Fractions — Further open questions

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $K$ be a degree-three extension of $\mathbb Q$ all of whose embeddings into $\mathbb C$ have image in $\mathbb R$, and fix one real embedding to compare its elements with $0$ and $1$. Let $y,z\in K$ satisfy $0<y,z<1$ and suppose that $1,y,z$ are linearly independent over $\mathbb Q$. Define the classical Jacobi--Perron iteration by $v_0=(x_0,y_0,z_0)=(1,y,z)$ and \[ a_i=\left\lfloor\frac{z_i}{y_i}\right\rfloor,\qquad b_i=\left\lfloor\frac{x_i}{y_i}\right\rfloor,\qquad (x_{i+1},y_{i+1},z_{i+1})=(y_i,z_i-a_i y_i,x_i-b_i y_i). \] Here $\lfloor u\rfloor$ denotes the greatest integer at most $u$. Each step is an invertible integer linear transformation, so linear independence is preserved and none of the coordinates vanishes. The output is the sequence of integer pairs $(a_i,b_i)$. It is eventually periodic if there are integers $N\ge0$ and $p\ge1$ such that $(a_{i+p},b_{i+p})=(a_i,b_i)$ for all $i\ge N$. [S3]

Statement recorded in the supplied source.

For every such field $K$ and every such pair $y,z$, is this Jacobi--Perron output eventually periodic? [S2][S3]

Short English statement

Does the classical Jacobi--Perron algorithm eventually repeat for every independent basis triple of a totally real cubic field with the two nonconstant entries between zero and one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1900020 (AMR-018-0020). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Oleg Karpenkov, Open problems in geometry of continued fractions, 2017. Problem 20 and its surrounding definitions and remarks. https://arxiv.org/pdf/1712.01450.
• S3 Oleg Karpenkov, On Hermite's problem, Jacobi--Perron type algorithms, and Dirichlet groups, arXiv:2101.12707, 2021. Section 2.1, pp. 5--7, algorithm and Problem 1 (Jacobi's Last Theorem). https://arxiv.org/pdf/2101.12707.