Bugeaud Collection of Conjectures and Open Questions: p-adic Littlewood Conjecture

Problem 10.8 (p-adic Littlewood conjecture). For every real number ξ and every prime number p, inf_q ≥ 1 q · lVert q ξ rVert · |q|_p = 0, where lVert · rVert denotes the distance to the nearest integer and |·|_p denotes the p-adic absolute value. Posed by de Mathan and Teulié [dMT04].

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-book-bugeaud-distribution-modulo-one-problem10-8,
  title        = {Bugeaud Collection of Conjectures and Open Questions: p-adic Littlewood Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/book-bugeaud-distribution-modulo-one-problem10-8}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Problem 10.8 ( $p$ -adic Littlewood conjecture). For every real number $ξ$ and every prime number $p$ , $q \geq 1 in f q \cdot ∥ q ξ ∥ \cdot ∣ q ∣_{p} = 0,$ where $∥ \cdot ∥$ denotes the distance to the nearest integer and $∣ \cdot ∣_{p}$ denotes the $p$ -adic absolute value. Posed by de Mathan and Teulié [dMT04].

This is the $p$ -adic analogue of the Littlewood conjecture, posed by de Mathan and Teulié. A liminf-based formulation also appears in the file FormalConjectures/Wikipedia/LittlewoodConjecture.lean as padic_littlewood_conjecture.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_8.

theorem problem_10_8 (ξ : ℝ) (p : ℕ) (hp : p.Prime) :
    sInf {x : ℝ | ∃ q : ℕ, 1 ≤ q ∧
      x = q * padicNorm p q * distToNearestInt (q * ξ)} = 0

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.
[dMT04] de Mathan, Bernard, and Olivier Teulié. "Problèmes diophantiens simultanés." Monatshefte für Mathematik 143.3 (2004): 229-245.
[EK07] Einsiedler, Manfred, and Dmitry Kleinbock. "Measure rigidity and $p$ -adic Littlewood-type problems." Compositio Mathematica 143.3 (2007): 689-702.

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.