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import FormalConjecturesUtil
import FormalConjecturesForMathlib.Data.Real.NearestIntBugeaud 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.
[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 := ξ:ℝp:ℕhp:Nat.Prime p⊢ sInf {x | ∃ q, 1 ≤ q ∧ x = ↑q * ↑(padicNorm p ↑q) * distToNearestInt (↑q * ξ)} = 0
All goals completed! 🐙
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 := ξ:ℝp:ℕhp:Nat.Prime phξ:(minpoly ℚ ξ).natDegree = 2⊢ sInf {x | ∃ q, 1 ≤ q ∧ x = ↑q * ↑(padicNorm p ↑q) * distToNearestInt (↑q * ξ)} = 0
All goals completed! 🐙
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 := p:ℕhp:Nat.Prime p⊢ dimH {ξ | Filter.liminf (fun q => ↑q * ↑(padicNorm p ↑q) * distToNearestInt (↑q * ξ)) Filter.atTop ≠ 0} = 0
All goals completed! 🐙
end Bugeaud08