Bounded continued fractions of cubic algebraic irrational coordinates

Determine whether the cubic irrational numbers \(\alpha_j=a_j\theta+b_j\theta^2\), arising from the cubic-field Kronecker construction, have bounded continued-fraction partial quotients and therefore whether their one-dimensional Kronecker projections are quasi-uniform; in particular, resolve the conjecture that every real algebraic irrational of degree at least three has unbounded partial quotients.

Background

The paper proves uniform quasi-uniformity for every two-dimensional coordinate projection of the cubic-field Kronecker sequence, but explicitly distinguishes this result from the corresponding one-dimensional question. For a one-dimensional Kronecker sequence, quasi-uniformity is equivalent to bounded continued-fraction partial quotients of its generating number.

Each coordinate $\alpha_j=a_j\theta+b_j\theta^2$ is a cubic irrational. The paper records the widely held conjecture that real algebraic irrationals of degree at least three have unbounded partial quotients. If this conjecture is true, then none of the one-dimensional projections of the construction would be quasi-uniform, despite the established quasi-uniformity of all bivariate projections.