FormalConjecturesForMathlib.NumberTheory.Lacunary

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module


public import Mathlib.Analysis.Normed.Field.Lemmas
public import Mathlib.Order.Filter.Defs
import Mathlib.Tactic.Rify

@[expose] public sectionopen Filter

Say a sequence is lacunary if there exists some $\lambda > 1$ such that $a_{n+1}/a_n > \lambda$ for all sufficiently large $n$.

def IsLacunary (n : ℕ → ℕ) : Prop := ∃ c > (1 : ℝ), ∀ᶠ k in atTop, c * n k < n (k + 1)

Every lacunary sequence is eventually strictly increasing.

lemma IsLacunary.eventually_lt {n : ℕ → ℕ} (hn : IsLacunary n) : ∀ᶠ k in atTop, n k < n (k + 1) := byn:ℕ → ℕhn:IsLacunary n⊢ ∀ᶠ (k : ℕ) in atTop, n k < n (k + 1)
  obtain ⟨c, hc, h⟩ := hnn:ℕ → ℕc:ℝhc:c > 1h:∀ᶠ (k : ℕ) in atTop, c * ↑(n k) < ↑(n (k + 1))⊢ ∀ᶠ (k : ℕ) in atTop, n k < n (k + 1)
  obtain ⟨N, h⟩ := eventually_atTop.1 hn:ℕ → ℕc:ℝhc:c > 1h✝:∀ᶠ (k : ℕ) in atTop, c * ↑(n k) < ↑(n (k + 1))N:ℕh:∀ (b : ℕ), N ≤ b → c * ↑(n b) < ↑(n (b + 1))⊢ ∀ᶠ (k : ℕ) in atTop, n k < n (k + 1)
  refine eventually_atTop.2 ⟨N, fun b hb ↦ ?_⟩n:ℕ → ℕc:ℝhc:c > 1h✝:∀ᶠ (k : ℕ) in atTop, c * ↑(n k) < ↑(n (k + 1))N:ℕh:∀ (b : ℕ), N ≤ b → c * ↑(n b) < ↑(n (b + 1))b:ℕhb:N ≤ b⊢ n b < n (b + 1)
  grw [hcn:ℕ → ℕc:ℝhc:c > 1h✝:∀ᶠ (k : ℕ) in atTop, c * ↑(n k) < ↑(n (k + 1))N:ℕh:∀ (b : ℕ), N ≤ b → 1 * ↑(n b) < ↑(n (b + 1))b:ℕhb:N ≤ b⊢ n b < n (b + 1)]n:ℕ → ℕc:ℝhc:c > 1h✝:∀ᶠ (k : ℕ) in atTop, c * ↑(n k) < ↑(n (k + 1))N:ℕh:∀ (b : ℕ), N ≤ b → 1 * ↑(n b) < ↑(n (b + 1))b:ℕhb:N ≤ b⊢ n b < n (b + 1) at h
  simpa using h _ hbAll goals completed! 🐙

The real-valued analogue of IsLacunary.

def IsLacunaryReal (a : ℕ → ℝ) : Prop :=
  ∃ c > (1 : ℝ), ∀ᶠ k in atTop, c * a k < a (k + 1)

An ℕ-valued sequence is lacunary iff its real cast is lacunary as a real sequence.

lemma isLacunary_iff_isLacunaryReal {n : ℕ → ℕ} :
    IsLacunary n ↔ IsLacunaryReal (fun k => (n k : ℝ)) := Iff.rfl

A positive lacunary real-valued sequence is eventually strictly increasing.

lemma IsLacunaryReal.eventually_lt {a : ℕ → ℝ} (ha : IsLacunaryReal a)
    (hpos : ∀ᶠ k in atTop, 0 < a k) : ∀ᶠ k in atTop, a k < a (k + 1) := bya:ℕ → ℝha:IsLacunaryReal ahpos:∀ᶠ (k : ℕ) in atTop, 0 < a k⊢ ∀ᶠ (k : ℕ) in atTop, a k < a (k + 1)
  obtain ⟨c, hc, hlac⟩ := haa:ℕ → ℝhpos:∀ᶠ (k : ℕ) in atTop, 0 < a kc:ℝhc:c > 1hlac:∀ᶠ (k : ℕ) in atTop, c * a k < a (k + 1)⊢ ∀ᶠ (k : ℕ) in atTop, a k < a (k + 1)
  filter_upwards [hlac, hpos] with k hka:ℕ → ℝhpos:∀ᶠ (k : ℕ) in atTop, 0 < a kc:ℝhc:c > 1hlac:∀ᶠ (k : ℕ) in atTop, c * a k < a (k + 1)k:ℕhk:c * a k < a (k + 1)⊢ 0 < a k → a k < a (k + 1) hpa:ℕ → ℝhpos:∀ᶠ (k : ℕ) in atTop, 0 < a kc:ℝhc:c > 1hlac:∀ᶠ (k : ℕ) in atTop, c * a k < a (k + 1)k:ℕhk:c * a k < a (k + 1)hp:0 < a k⊢ a k < a (k + 1) using bya:ℕ → ℝhpos:∀ᶠ (k : ℕ) in atTop, 0 < a kc:ℝhc:c > 1hlac:∀ᶠ (k : ℕ) in atTop, c * a k < a (k + 1)k:ℕhk:c * a k < a (k + 1)hp:0 < a k⊢ a k < a (k + 1) nlinarithAll goals completed! 🐙

def IsSublacunary (m : ℕ → ℕ) : Prop :=
  Filter.Tendsto (fun n => (m (n + 1) : ℝ) / m n) Filter.atTop (nhds 1)