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module
public import FormalConjecturesForMathlib.Topology.Algebra.InfiniteSum.Group
public import FormalConjecturesForMathlib.Topology.Algebra.InfiniteSum.Order
public import Mathlib.Analysis.SpecificLimits.Basic@[expose] public sectionSequences 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.
open Set Filter Topologyopen scoped Finset
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.
def HasFabryGaps (n : ℕ → ℕ) : Prop := StrictMono n ∧ Tendsto (fun k => n k / (k : ℝ)) atTop atTop
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.
def HasFejerGaps (n : ℕ → ℕ) : Prop := StrictMono n ∧ Summable (fun k => (n k : ℝ)⁻¹)f:ℕ → ℕhf:HasFejerGaps fN:ℕhN:1 < Nthis✝:0 < f Nn:ℕhn:N / 2 ≤ n ∧ n < Nthis:0 < f n⊢ (↑(f N))⁻¹ ≤ (↑(f n))⁻¹
grw [f:ℕ → ℕhf:HasFejerGaps fN:ℕhN:1 < Nthis✝:0 < f Nn:ℕhn:N / 2 ≤ n ∧ n < Nthis:0 < f n⊢ (↑(f N))⁻¹ ≤ (↑(f N))⁻¹All goals completed! 🐙
_ ≤ ∑' n : Ici (N / 2), (f n : ℝ)⁻¹ :=
hf.2.sum_le_tsum_set (f:ℕ → ℕhf:HasFejerGaps fN:ℕhN:1 < Nthis:0 < f N⊢ ↑(Finset.Ico (N / 2) N) ⊆ Ici (N / 2) All goals completed! 🐙) (f:ℕ → ℕhf:HasFejerGaps fN:ℕhN:1 < Nthis:0 < f N⊢ ∀ i ∉ Finset.Ico (N / 2) N, i ∈ Ici (N / 2) → 0 ≤ (↑(f i))⁻¹ All goals completed! 🐙)
f:ℕ → ℕhf:HasFejerGaps f⊢ Tendsto (fun x ↦ ∑' (n : ↑(Ici x)), (↑(f ↑n))⁻¹) atTop (𝓝 0) exact hf.2.tendsto_tsum_zero (fun b ↦ f:ℕ → ℕhf:HasFejerGaps fb:ℕ⊢ ∀ᶠ (i : ℕ) in atTop, b ∉ Ici i All goals completed! 🐙)
f:ℕ → ℕhf:HasFejerGaps f⊢ ∀ᶠ (n : ℕ) in atTop, ∑' (n_1 : ↑(Ici n)), (↑(f ↑n_1))⁻¹ ∈ Ioi 0 filter_upwards [eventually_gt_atTop 0] with N f:ℕ → ℕhf:HasFejerGaps fN:ℕhN:0 < N⊢ ∑' (n : ↑(Ici N)), (↑(f ↑n))⁻¹ ∈ Ioi 0
refine (hf.2.comp_injective Subtype.val_injective).tsum_pos (f:ℕ → ℕhf:HasFejerGaps fN:ℕhN:0 < N⊢ ∀ (i : { x // x ∈ Ici N }), 0 ≤ ((fun k ↦ (↑(f k))⁻¹) ∘ Subtype.val) i All goals completed! 🐙) ⟨N, f:ℕ → ℕhf:HasFejerGaps fN:ℕhN:0 < N⊢ N ∈ Ici N All goals completed! 🐙⟩ ?_
All goals completed! 🐙