Bugeaud Collection of Conjectures and Open Questions: Pisot orbits on the Cantor set

The question

Problem 10.61. Let $\alpha >2$ be a Pisot number. For every $\xi \in C(\alpha )$ the sequence $(\xi {\alpha }^n{)}_{n\ge 1}$ is not uniformly distributed modulo one.

Problem 10.61, proposed by Mendès France [MF67, Problème 1, p. 41]. For a Pisot number $\alpha >2$ put $$C(\alpha )=\left\{(\alpha -1)\sum _{k\ge 1}{\epsilon }_k{\alpha }^{-k}:{\epsilon }_k\in \{0,1\}\right\}.$$ The problem asks to prove that $(\xi {\alpha }^n{)}_{n\ge 1}$ is not uniformly distributed modulo one for any $\xi \in C(\alpha )$. In the language of [MF67] this reads $C(\alpha )\cap B(\alpha )=\varnothing$, where $B(\alpha )$ is the set of $\xi$ for which $(\xi {\alpha }^n)$ is uniformly distributed modulo one.

The problem is open. It is easy for an integer base and settled for two quadratic $\alpha$; the general Pisot case is what remains. [Ste26] reduces it to a statement about shift-invariant measures of the full two-shift, and proves that every true instance admits a finite certificate, so the difficulty is uniformity in $\alpha$ rather than any single $\alpha$. The variants below state the criterion and the instances of [Ste26] that concern Problem 10.61 directly; the reduction itself is not stated here, since it needs the symbolic model.

Deleting the term $n=0$ does not change uniform distribution modulo one, so the statements indexed from $n=1$ agree with the ones of [Ste26], which are indexed from $n=0$. Where an orbit is confined away from an interval the stronger form, for every $n\ge 0$, is stated.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_61.

theorem problem_10_61 (α : ℝ) (hα : IsPisot α) (hα2 : 2 < α) :
    ∀ ξ ∈ pisotCantorSet α, ¬ IsEquidistributedModuloOne fun n : ℕ => ξ * α ^ (n + 1)

What counts as progress

• A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• [Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Vol. 193. Cambridge University Press, 2012. Chapter 10, Problem 10.61, p. 222.
• [MF67] Mendès France, Michel. "Nombres normaux. Applications aux fonctions pseudo-aléatoires." Journal d'Analyse Mathématique 20 (1967): 1-56.
• [Kok35] Koksma, Jurjen F. "Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins." Compositio Mathematica 2 (1935): 250-258.
• [Ste26] Stephan, Ralf. "Criteria for the non-equidistribution of $(\xi {\alpha }^n)$ on the Cantor set $C(\alpha )$." Preprint, 2026. https://doi.org/10.13140/RG.2.2.13923.52001

Source and licence

Imported from Formal Conjectures (books), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.