Problem10_5.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Bugeaud Collection of Conjectures and Open Questions: Lacunary Sequences in Real Number Fields

The following problems were proposed and discussed by Dubickas as Conjecture 2 in [Dub09].

References:

[Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Vol. 193. Cambridge University Press, 2012. Chapter 10.
[Dub09] Dubickas, Artūras. "An approximation property of lacunary sequences." Israel Journal of Mathematics 170.1 (2009): 95-111.

Imports (1)

FormalConjecturesUtil

theorem Bugeaud05.problem_10_5 sorry line 53 source
depends on sorry what this means
Statement uses: IntermediateField, Rat, Real, FiniteDimensional, Subtype, Membership.mem, LT.lt, OfNat.ofNat, Exists, Nat, And, IsLacunaryReal, Not, LE.le, HSub.hSub, Filter.limsup, Int.fract, HMul.hMul, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
113 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Bugeaud05.problem_10_5_moreover sorry line 81 source
depends on sorry what this means
Statement uses: IntermediateField, Rat, Real, FiniteDimensional, Subtype, Membership.mem, LT.lt, OfNat.ofNat, Exists, Nat, And, IsLacunaryReal, Not, Set, Set.Icc, HSub.hSub, HAdd.hAdd, MapClusterPt, Filter.atTop, Int.fract and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
121 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Bugeaud05.problem_10_5_of_moreover The "moreover" form of Problem 10.5 implies the first part: applying the cluster-point density to the subinterval line 94 source
Source documentation
The "moreover" form of Problem 10.5 implies the first part: applying the cluster-point density to the subinterval $[1 - \varepsilon, 1]$ yields the required lower bound on the limsup.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IntermediateField, Rat, Real, FiniteDimensional, Subtype, Membership.mem, LT.lt, OfNat.ofNat, Exists, Nat, And, IsLacunaryReal, Not, Set, Set.Icc, HSub.hSub, HAdd.hAdd, MapClusterPt, Filter.atTop, Int.fract and 3 more
Proof uses: Exists.intro, And.intro, Filter.IsBoundedUnder, Filter.isBoundedUnder_of, LT.lt.le, Int.fract_lt_one, dite, le_of_not_gt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance, Ring, Int.negOfNat, HPow.hPow, Nat.rawCast, Mathlib.Tactic.Ring.Common.sub_congr and 53 more
191 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
```
theorem problem_10_5_of_moreover (h : type_of% problem_10_5_moreover) :
    type_of% problem_10_5 := by
…
```
Mentions: Bugeaud05.problem_10_5_moreover, Bugeaud05.problem_10_5