5 problems

Effective Lang conjecture. There exists an absolute constant $C$ such that, for every real cubic irrational number $x$, all partial quotients satisfy

A torus decomposition arising from a periodic two-dimensional continued fraction of cubic irrationality consists of faces with integer affine types. Triangular-face conjecture. For…

A torus decomposition arising from a periodic two-dimensional continued fraction of cubic irrationality has faces with integer distances to the origin. Unit-face-distance conjectur…

A torus decomposition arising from a periodic two-dimensional continued fraction of cubic irrationality has faces, each equipped with an integer distance to the origin. Face-distan…

Let $\xi\in\mathbb{R}$ be a real cubic irrational number, let $H(\xi)$ be the naive height of its minimal polynomial with coprime integer coefficients, and let…