/-- **Gallagher's ergodic theorem** on Diophantine approximation. -/
theorem addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) :
    (∀ᵐ x, ¬addWellApproximable 𝕊 δ x) ∨ ∀ᵐ x, addWellApproximable 𝕊 δ x := by
  /- Sketch of proof:

    Let `E := addWellApproximable 𝕊 δ`. For each prime `p : ℕ`, we can partition `E` into three
    pieces `E = (A p) ∪ (B p) ∪ (C p)` where:
      `A p = blimsup (approxAddOrderOf 𝕊 n (δ n)) atTop (fun n => 0 < n ∧ (p ∤ n))`
      `B p = blimsup (approxAddOrderOf 𝕊 n (δ n)) atTop (fun n => 0 < n ∧ (p ∣∣ n))`
      `C p = blimsup (approxAddOrderOf 𝕊 n (δ n)) atTop (fun n => 0 < n ∧ (p*p ∣ n))`.
    In other words, `A p` is the set of points `x` for which there exist infinitely-many `n` such
    that `x` is within a distance `δ n` of a point of order `n` and `p ∤ n`. Similarly for `B`, `C`.

    These sets have the following key properties:
      1. `A p` is almost invariant under the ergodic map `y ↦ p • y`
      2. `B p` is almost invariant under the ergodic map `y ↦ p • y + 1/p`
      3. `C p` is invariant under the map `y ↦ y + 1/p`
    To prove 1 and 2 we need the key result `blimsup_thickening_mul_ae_eq` but 3 is elementary.

    It follows from `AddCircle.ergodic_nsmul_add` and `Ergodic.ae_empty_or_univ_of_image_ae_le` that
    if either `A p` or `B p` is not almost empty for any `p`, then it is almost full and thus so is
    `E`. We may therefore assume that `A p` and `B p` are almost empty for all `p`. We thus have
    `E` is almost equal to `C p` for every prime. Combining this with 3 we find that `E` is almost
    invariant under the map `y ↦ y + 1/p` for every prime `p`. The required result then follows from
    `AddCircle.ae_empty_or_univ_of_forall_vadd_ae_eq_self`. -/
  letI : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup _
  set μ : Measure 𝕊 := volume
  set u : Nat.Primes → 𝕊 := fun p => ↑((↑(1 : ℕ) : ℝ) / ((p : ℕ) : ℝ) * T)
  have hu₀ : ∀ p : Nat.Primes, addOrderOf (u p) = (p : ℕ) := by
    rintro ⟨p, hp⟩; exact addOrderOf_div_of_gcd_eq_one hp.pos (gcd_one_left p)
  have hu : Tendsto (addOrderOf ∘ u) atTop atTop := by
    rw [(funext hu₀ : addOrderOf ∘ u = (↑))]
    have h_mono : Monotone ((↑) : Nat.Primes → ℕ) := fun p q hpq => hpq
    refine' h_mono.tendsto_atTop_atTop fun n => _
    obtain ⟨p, hp, hp'⟩ := n.exists_infinite_primes
    exact ⟨⟨p, hp'⟩, hp⟩
  set E := addWellApproximable 𝕊 δ
  set X : ℕ → Set 𝕊 := fun n => approxAddOrderOf 𝕊 n (δ n)
  set A : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p∤n
  set B : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p∣∣n
  set C : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ^ 2 ∣ n
  have hA₀ : ∀ p, MeasurableSet (A p) := fun p =>
    MeasurableSet.measurableSet_blimsup fun n _ => isOpen_thickening.measurableSet
  have hB₀ : ∀ p, MeasurableSet (B p) := fun p =>
    MeasurableSet.measurableSet_blimsup fun n _ => isOpen_thickening.measurableSet
  have hE₀ : NullMeasurableSet E μ := by
    refine' (MeasurableSet.measurableSet_blimsup fun n hn =>
      IsOpen.measurableSet _).nullMeasurableSet
    

Proof state:
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p => ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n => approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
n : ℕ
hn : 0 < n
⊢ IsOpen (approxAddOrderOf 𝕊 n (δ n))

Next tactic:
exact isOpen_thickening