/-- **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
    exact isOpen_thickening
  have hE₁ : ∀ p, E = A p ∪ B p ∪ C p := by
    intro p
    simp only [addWellApproximable, ← blimsup_or_eq_sup, ← and_or_left, ← sup_eq_union, sq]
    congr
    ext n
    tauto
  have hE₂ : ∀ p : Nat.Primes, A p =ᵐ[μ] (∅ : Set 𝕊) ∧ B p =ᵐ[μ] (∅ : Set 𝕊) → E =ᵐ[μ] C p := by
    rintro p ⟨hA, hB⟩
    rw [hE₁ p]
    exact union_ae_eq_right_of_ae_eq_empty ((union_ae_eq_right_of_ae_eq_empty hA).trans hB)
  have hA : ∀ p : Nat.Primes, A p =ᵐ[μ] (∅ : Set 𝕊) ∨ A p =ᵐ[μ] univ := by
    rintro ⟨p, hp⟩
    let f : 𝕊 → 𝕊 := fun y => (p : ℕ) • y
    suffices
      f '' A p ⊆ blimsup (fun n => approxAddOrderOf 𝕊 n (p * δ n)) atTop fun n => 0 < n ∧ p∤n by
      apply (ergodic_nsmul hp.one_lt).ae_empty_or_univ_of_image_ae_le (hA₀ p)
      apply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl
      exact blimsup_thickening_mul_ae_eq μ (fun n => 0 < n ∧ p∤n) (fun n => {y | addOrderOf y = n})
        (Nat.cast_pos.mpr hp.pos) _ hδ
    refine' (SupHom.apply_blimsup_le (sSupHom.setImage f)).trans (mono_blimsup fun n hn => _)
    replace hn := Nat.coprime_comm.mp (hp.coprime_iff_not_dvd.2 hn.2)
    exact approxAddOrderOf.image_nsmul_subset_of_coprime (δ n) hp.pos hn
  have hB : ∀ p : Nat.Primes, B p =ᵐ[μ] (∅ : Set 𝕊) ∨ B p =ᵐ[μ] univ := by
    rintro ⟨p, hp⟩
    let x := u ⟨p, hp⟩
    let f : 𝕊 → 𝕊 := fun y => p • y + x
    suffices
      f '' B p ⊆ blimsup (fun n => approxAddOrderOf 𝕊 n (p * δ n)) atTop fun n => 0 < n ∧ p∣∣n by
      apply (ergodic_nsmul_add x hp.one_lt).ae_empty_or_univ_of_image_ae_le (hB₀ p)
      apply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl
      exact blimsup_thickening_mul_ae_eq μ (fun n => 0 < n ∧ p∣∣n) (fun n => {y | addOrderOf y = n})
        (Nat.cast_pos.mpr hp.pos) _ hδ
    refine' (SupHom.apply_blimsup_le (sSupHom.setImage f)).trans (mono_blimsup _)
    rintro n ⟨hn, h_div, h_ndiv⟩
    have h_cop : (addOrderOf x).Coprime (n / p) := by
      obtain ⟨q, rfl⟩ := h_div
      rw [hu₀, Subtype.coe_mk, hp.coprime_iff_not_dvd, q.mul_div_cancel_left hp.pos]
      exact fun contra => h_ndiv (mul_dvd_mul_left p contra)
    replace h_div : n / p * p = n := Nat.div_mul_cancel h_div
    have hf : f = (fun y => x + y) ∘ fun y => p • y := by ext; simp [add_comm x]; ac_rfl
    simp only at hf
    simp_rw [Function.comp_apply, le_eq_subset]
    rw [sSupHom.setImage_toFun, hf, image_comp]
    

Proof state:
case mk.intro.intro
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)
hE₀ : NullMeasurableSet E
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ
p : ℕ
hp : Nat.Prime p
x : 𝕊 := u { val := p, property := hp }
f : 𝕊 → 𝕊 := fun y => p • y + x
n : ℕ
hn : 0 < n
h_ndiv : ¬p * p ∣ n
h_cop : Nat.Coprime (addOrderOf x) (n / p)
h_div : n / p * p = n
hf : (fun y => p • y + ↑(↑1 / ↑p * T)) = (fun y => ↑(↑1 / ↑p * T) + y) ∘ fun y => p • y
⊢ (fun y => ↑(↑1 / ↑p * T) + y) '' ((fun y => p • y) '' approxAddOrderOf 𝕊 n (δ n)) ⊆ approxAddOrderOf 𝕊 n (↑p * δ n)

Next tactic:
have := @monotone_image 𝕊 𝕊 fun y => x + y