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

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

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

This event attributed to Fable 5

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

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

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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. parent of this event

This event attributed to Opus 5

This event attributed to Ralf Stephan