$(\xi\alpha^n)_{n\ge1}$ is not uniformly distributed modulo one for Pisot $\alpha$ and $\xi$ in the Cantor set $C(\alpha)$

Bugeaud's Problem 10.61, due to Michel Mendès France in 1967: for a Pisot number $\alpha > 2$ and the Cantor set $C(\alpha) = \{(\alpha-1)\sum_{k\ge1}\varepsilon_k\alpha^{-k} : \varepsilon_k \in \{0,1\}\}$, no $\xi \in C(\alpha)$ has $(\xi\alpha^n)_{n\ge1}$ uniformly distributed modulo one. The problem itself remains open. What is proved is a set of criteria for it, and two instances. The criteria: a reduction to symbolic dynamics that is an equivalence; a pressure criterion; and a covering criterion which, for a quadratic setup of norm $b$, applies exactly when $(\log_2\alpha - 1)(\log_2(\alpha/|b|) - 1) > 1$, a condition that reduces to $\alpha > 4$ for units. The two instances are both quadratic: at $\alpha = 2+\sqrt5$ in the strong form, an explicit interval that every orbit misses at every time, and at $\alpha = 2+\sqrt3$ by a confinement-gap certificate.

The problem is open, and the repository says so: what is proved are criteria for it and two instances, not the general case. Every compared statement that concludes Problem 10.61 does so for a quadratic setup - a real root $\alpha > 1$ of $X^2 - aX - b$ whose conjugate has modulus below one - and both instances are quadratic. The arbitrary-degree material is conditional ingredients: for the family $X^d - aX^{d-1} - 1$ the real root exceeding $a$ is shown to be Pisot for $a \ge 3$, with a conjugate-modulus bound and a numerical inequality. No compared statement carries those above degree two, because the covering criterion is proved only for quadratic setups. The covering criterion also leaves quadratic $\alpha$ with route-A exponent at least one undecided, about which nothing is claimed.