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

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import FormalConjecturesUtil

/-!
# Bugeaud Collection of Conjectures and Open Questions: $p$-adic Littlewood Conjecture

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`.

*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.
-/

namespace Bugeaud08

/--
Problem 10.8 ($p$-adic Littlewood conjecture). For every real number $\xi$ and
every prime number $p$,
$$\inf_{q \ge 1} q \cdot \lVert q \xi \rVert \cdot |q|_p = 0,$$
where $\lVert \cdot \rVert$ denotes the distance to the nearest integer and
$|\cdot|_p$ denotes the $p$-adic absolute value. Posed by de Mathan and
Teulié [dMT04].
-/
@[category research open, AMS 11]
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 := by
  sorry

/--
The quadratic case of Problem 10.8. de Mathan and Teulié [dMT04] solved the
$p$-adic Littlewood conjecture for quadratic real numbers.
-/
@[category research solved, AMS 11]
theorem problem_10_8.variants.quadratic (ξ : ℝ) (p : ℕ) (hp : p.Prime)
    (hξ : (minpoly ℚ ξ).natDegree = 2) :
    sInf {x : ℝ | ∃ q : ℕ, 1 ≤ q ∧
      x = q * padicNorm p q * distToNearestInt (q * ξ)} = 0 := by
  sorry

/--
The exceptional set for Problem 10.8 has Hausdorff dimension zero (Einsiedler
and Kleinbock [EK07].
-/
@[category research solved, AMS 11]
theorem problem_10_8.variants.exceptional_set_dimH_zero (p : ℕ) (hp : p.Prime) :
    dimH {ξ : ℝ | Filter.atTop.liminf
      (fun q : ℕ ↦ q * padicNorm p q * distToNearestInt (q * ξ)) ≠ 0} = 0 := by
  sorry

end Bugeaud08

• Formalization source ↗

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