/-- If a null-measurable subset of the circle is almost invariant under rotation by a family of
rational angles with denominators tending to infinity, then it must be almost empty or almost full.
-/
theorem ae_empty_or_univ_of_forall_vadd_ae_eq_self {s : Set <| AddCircle T}
    (hs : NullMeasurableSet s volume) {ι : Type*} {l : Filter ι} [l.NeBot] {u : ι → AddCircle T}
    (hu₁ : ∀ i, (u i +ᵥ s : Set _) =ᵐ[volume] s) (hu₂ : Tendsto (addOrderOf ∘ u) l atTop) :
    s =ᵐ[volume] (∅ : Set <| AddCircle T) ∨ s =ᵐ[volume] univ := by
  /- Sketch of proof:
    Assume `T = 1` for simplicity and let `μ` be the Haar measure. We may assume `s` has positive
    measure since otherwise there is nothing to prove. In this case, by Lebesgue's density theorem,
    there exists a point `d` of positive density. Let `Iⱼ` be the sequence of closed balls about `d`
    of diameter `1 / nⱼ` where `nⱼ` is the additive order of `uⱼ`. Since `d` has positive density we
    must have `μ (s ∩ Iⱼ) / μ Iⱼ → 1` along `l`. However since `s` is invariant under the action of
    `uⱼ` and since `Iⱼ` is a fundamental domain for this action, we must have
    `μ (s ∩ Iⱼ) = nⱼ * μ s = (μ Iⱼ) * μ s`. We thus have `μ s → 1` and thus `μ s = 1`. -/
  set μ := (volume : Measure <| AddCircle T)
  set n : ι → ℕ := addOrderOf ∘ u
  have hT₀ : 0 < T := hT.out
  have hT₁ : ENNReal.ofReal T ≠ 0 := by simpa
  rw [ae_eq_empty, ae_eq_univ_iff_measure_eq hs, AddCircle.measure_univ]
  rcases eq_or_ne (μ s) 0 with h | h; · exact Or.inl h
  right
  obtain ⟨d, -, hd⟩ : ∃ d, d ∈ s ∧ ∀ {ι'} {l : Filter ι'} (w : ι' → AddCircle T) (δ : ι' → ℝ),
    Tendsto δ l (𝓝[>] 0) → (∀ᶠ j in l, d ∈ closedBall (w j) (1 * δ j)) →
      Tendsto (fun j => μ (s ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1) :=
    exists_mem_of_measure_ne_zero_of_ae h
      (IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div μ s 1)
  let I : ι → Set (AddCircle T) := fun j => closedBall d (T / (2 * ↑(n j)))
  replace hd : Tendsto (fun j => μ (s ∩ I j) / μ (I j)) l (𝓝 1)
  · let δ : ι → ℝ := fun j => T / (2 * ↑(n j))
    have hδ₀ : ∀ᶠ j in l, 0 < δ j :=
      (hu₂.eventually_gt_atTop 0).mono fun j hj => div_pos hT₀ <| by positivity
    have hδ₁ : Tendsto δ l (𝓝[>] 0) := by
      refine' tendsto_nhdsWithin_iff.mpr ⟨_, hδ₀⟩
      replace hu₂ : Tendsto (fun j => T⁻¹ * 2 * n j) l atTop :=
        (tendsto_nat_cast_atTop_iff.mpr hu₂).const_mul_atTop (by positivity : 0 < T⁻¹ * 2)
      convert hu₂.inv_tendsto_atTop
      ext j
      simp only [Pi.inv_apply, mul_inv_rev, inv_inv, div_eq_inv_mul, ← mul_assoc]
    have hw : ∀ᶠ j in l, d ∈ closedBall d (1 * δ j) := hδ₀.mono fun j hj => by
      simp only [comp_apply, one_mul, mem_closedBall, dist_self]
      apply hj.le
    exact hd _ δ hδ₁ hw
  suffices ∀ᶠ j in l, μ (s ∩ I j) / μ (I j) = μ s / ENNReal.ofReal T by
    replace hd := hd.congr' this
    rwa [tendsto_const_nhds_iff, ENNReal.div_eq_one_iff hT₁ ENNReal.ofReal_ne_top] at hd
  refine' (hu₂.eventually_gt_atTop 0).mono fun j hj => _
  have : addOrderOf (u j) = n j := rfl
  have huj : IsOfFinAddOrder (u j) := addOrderOf_pos_iff.mp hj
  have huj' : 1 ≤ (↑(n j) : ℝ) := by norm_cast
  have hI₀ : μ (I j) ≠ 0 := (measure_closedBall_pos _ d <| by positivity).ne.symm
  have hI₁ : μ (I j) ≠ ⊤ := measure_ne_top _ _
  have hI₂ : μ (I j) * ↑(n j) = ENNReal.ofReal T := by
    rw [volume_closedBall, mul_div, mul_div_mul_left T _ two_ne_zero,
      min_eq_right (div_le_self hT₀.le huj'), mul_comm, ← nsmul_eq_mul, ← ENNReal.ofReal_nsmul,
      nsmul_eq_mul, mul_div_cancel']
    exact Nat.cast_ne_zero.mpr hj.ne'
  

Proof state:
case inr.h.intro.intro
T : ℝ
hT : Fact (0 < T)
s : Set (AddCircle T)
ι : Type u_1
l : Filter ι
inst✝ : NeBot l
u : ι → AddCircle T
μ : Measure (AddCircle T) := volume
hs : NullMeasurableSet s
hu₁ : ∀ (i : ι), u i +ᵥ s =ᶠ[ae μ] s
n : ι → ℕ := addOrderOf ∘ u
hu₂ : Tendsto n l atTop
hT₀ : 0 < T
hT₁ : ENNReal.ofReal T ≠ 0
h : ↑↑μ s ≠ 0
d : AddCircle T
I : ι → Set (AddCircle T) := fun j => closedBall d (T / (2 * ↑(n j)))
hd : Tendsto (fun j => ↑↑μ (s ∩ I j) / ↑↑μ (I j)) l (𝓝 1)
j : ι
hj : 0 < n j
this : addOrderOf (u j) = n j
huj : IsOfFinAddOrder (u j)
huj' : 1 ≤ ↑(n j)
hI₀ : ↑↑μ (I j) ≠ 0
hI₁ : ↑↑μ (I j) ≠ ⊤
hI₂ : ↑↑μ (I j) * ↑(n j) = ENNReal.ofReal T
⊢ ↑↑μ (s ∩ I j) / ↑↑μ (I j) = ↑↑μ s / ENNReal.ofReal T

Next tactic:
rw [ENNReal.div_eq_div_iff hT₁ ENNReal.ofReal_ne_top hI₀ hI₁,
    volume_of_add_preimage_eq s _ (u j) d huj (hu₁ j) closedBall_ae_eq_ball, nsmul_eq_mul, ←
    mul_assoc, this, hI₂]