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(ξαn)n≥1(\xi\alpha^n)_{n\ge1} is not uniformly distributed modulo one for Pisot α\alpha and ξ\xi in the Cantor set C(α)C(\alpha)

Bugeaud's Problem 10.61, due to Michel Mendès France in 1967: for a Pisot number α>2\alpha > 2 and the Cantor set C(α)={(α−1)∑k≥1εkα−k:εk∈{0,1}}C(\alpha) = \{(\alpha-1)\sum_{k\ge1}\varepsilon_k\alpha^{-k} : \varepsilon_k \in \{0,1\}\}, no ξ∈C(α)\xi \in C(\alpha) has (ξαn)n≥1(\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 bb, applies exactly when (log⁡2α−1)(log⁡2(α/∣b∣)−1)>1(\log_2\alpha - 1)(\log_2(\alpha/|b|) - 1) > 1, a condition that reduces to α>4\alpha > 4 for units. The two instances are both quadratic: at α=2+5\alpha = 2+\sqrt5 in the strong form, an explicit interval that every orbit misses at every time, and at α=2+3\alpha = 2+\sqrt3 by a confinement-gap certificate.

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Computation
Field
Equidistribution mod 1
Posed by
Michel Mendès France
Year posed
1967
Years open
59y
Solved
2026-08-31
Model
Fable 5, Opus 5
Vendor
Anthropic
Collaborators
Ralf Stephan
Verification
Lean-checked, statement unaudited
Publication
Announced
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

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 α>1\alpha > 1 of X2−aX−bX^2 - aX - b whose conjugate has modulus below one - and both instances are quadratic. The arbitrary-degree material is conditional ingredients: for the family Xd−aXd−1−1X^d - aX^{d-1} - 1 the real root exceeding aa is shown to be Pisot for a≥3a \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 α\alpha with route-A exponent at least one undecided, about which nothing is claimed.

What the AI did

Directed by a human, the model planned and executed the discovery, the formalization and the write-up. The repository's formalization.yaml records the work as "(C) 2026 Ralf Stephan, in collaboration with Claude Code", and describes paper.pdf there as machine-written notes documenting the Lean development rather than a prior paper the formalization followed.

Verification

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 α=2+3\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.

Sources

Submitted by LucidKestrel185 on

Changelog2 changes

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