Source authenticated

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

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 31, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. Imported under CC BY 4.0.

Canonical aliases: $(\xi\alpha^n)_{n\ge1}$ is not uniformly distributed modulo one for Pisot $\alpha$ and $\xi$ in the Cantor set $C(\alpha)$ · Bugeaud Problem 10.61

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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Attribution

VibeMathed
registry · event recorded by

Fable 5
model · ai model contributor · Anthropic

Opus 5
model · ai model contributor · Anthropic

Ralf Stephan
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

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

specialist registry · passed

Lean-checked, statement unaudited. Registered in the Palomar registry as PALOMAR-2026-08-31-000013: status registered, trust high, pinned to commit d61132ff, mirrored to PalomarArchive. Seventeen statements are compared with leanprover/comparator, which checks three things per theorem - that the statement in Solution is definitionally the same statement as in the trusted Challenge module, compared constant by constant; that the proof uses no axiom outside a permitted list; and that the resulting environment is re-accepted by the Lean kernel. All seventeen registered theorems sit in the axiom-free lane, permitting only propext, Quot.sound and Classical.choice. The repository maintains a second lane permitting one cited literature input, LY.entropyRate_floor. Two theorems consume it and neither is among the registered seventeen; the $\alpha = 2+\sqrt3$ instance is registered in its axiom-free form. Not Lean-verified, on the anchoring half. A Palomar listing is a strong precondition rather than the anchoring itself: it makes the audit cheap, but nobody without a stake has checked that the formal statements say what Problem 10.61 says, and Palomar states plainly that a listing is not a certificate of novelty or relevance.

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Registry link

witness · pending

palomar link supplied by the VibeMathed registry.

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Github repo

code · pending

code link supplied by the VibeMathed registry.

Source ↗

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

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Artifacts and verifiers

Github repo

code · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

$(\xi\alpha^n)_{n\ge1}$ 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

Registry link evidence for this event

Github repo evidence for this event

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. parent of this event

This event attributed to Opus 5

This event attributed to Ralf Stephan

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