HasGaps.lean`

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

3 declarations, 0 contain sorry, 0 declared as axioms

Sequences with Large Gaps

References:

[Wa01] Wang, Yuefei. "On the Fatou set of an entire function with gaps." Tohoku Mathematical Journal, Second Series 53.1 (2001): 163-170.

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

def HasFabryGaps A sequence of natural numbers n₀ < n₁ < ... is said to have Fabry gaps if nₖ / k → ∞ line 37 source
Source documentation
A sequence of natural numbers n₀ < n₁ < ... is said to have Fabry gaps if nₖ / k → ∞. This is the terminology adopted in [Wa01] and some other sources.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, StrictMono, Filter.Tendsto, Real, HDiv.hDiv, Filter.atTop
7 instances, coercions and auxiliary constants left out.
```
def HasFabryGaps (n : ℕ → ℕ) : Prop := StrictMono n ∧ Tendsto (fun k => n k / (k : ℝ)) atTop atTop
```
def HasFejerGaps A sequence of natural numbers n₀ < n₁ < ... is said to have Fejér gaps if ∑' 1 / nₖ < ∞ line 41 source
Source documentation
A sequence of natural numbers n₀ < n₁ < ... is said to have Fejér gaps if ∑' 1 / nₖ < ∞. This is the terminology adopted in [Wa01] and some other sources.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, StrictMono, Summable, Real, Inv.inv, SummationFilter.unconditional
8 instances, coercions and auxiliary constants left out.
```
def HasFejerGaps (n : ℕ → ℕ) : Prop := StrictMono n ∧ Summable (fun k => (n k : ℝ)⁻¹)
```
theorem HasFejerGaps.hasFabryGaps line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, HasFejerGaps, HasFabryGaps
Proof uses: And.intro, StrictMono, Filter.Tendsto, Real, HDiv.hDiv, Filter.atTop, And.left, Summable, Inv.inv, SummationFilter.unconditional, Filter.tendsto_atTop_mono', Function.comp, OfNat.ofNat, tsum, Set.Ici, Membership.mem, Set, Filter.mp_mem, Set.ofPred, LT.lt and 79 more
155 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.