Problem10_8.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 3 contain sorry, 0 declared as axioms

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.

Imports (1)

FormalConjecturesUtil

theorem Bugeaud08.problem_10_8 Problem 10.8 ($p$-adic Littlewood conjecture) sorry line 49 source
Source documentation
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].

depends on sorry what this means
Statement uses: Real, Nat, Nat.Prime, Eq, InfSet.sInf, Set.ofPred, Exists, And, LE.le, OfNat.ofNat, HMul.hMul, padicNorm, Rat, distToNearestInt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
Mentions: distToNearestInt
theorem Bugeaud08.problem_10_8.variants.quadratic The quadratic case of Problem 10.8 sorry line 59 source
Source documentation
The quadratic case of Problem 10.8. de Mathan and Teulié [dMT04] solved the $p$-adic Littlewood conjecture for quadratic real numbers.

depends on sorry what this means
Statement uses: Real, Nat, Nat.Prime, Eq, Polynomial.natDegree, Rat, minpoly, OfNat.ofNat, InfSet.sInf, Set.ofPred, Exists, And, LE.le, HMul.hMul, padicNorm, distToNearestInt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
Mentions: distToNearestInt
theorem Bugeaud08.problem_10_8.variants.exceptional_set_dimH_zero The exceptional set for Problem 10.8 has Hausdorff dimension zero (Einsiedler and Kleinbock [EK07] sorry line 70 source
Source documentation
The exceptional set for Problem 10.8 has Hausdorff dimension zero (Einsiedler and Kleinbock [EK07].

depends on sorry what this means
Statement uses: Nat, Nat.Prime, Eq, ENNReal, dimH, Real, Set.ofPred, Ne, Filter.liminf, HMul.hMul, padicNorm, Rat, distToNearestInt, Filter.atTop, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
Mentions: distToNearestInt