File size: 2,533 Bytes
695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import Mathlib
namespace OAI
noncomputable section
open Set Filter MeasureTheory Metric
open scoped Topology ENNReal NNReal
open Set Filter MeasureTheory BoxIntegral
open scoped Topology
namespace YauCounterexamples
def middleBox (I : Box (Fin 3)) : Box (Fin 3) where
lower i := (3*I.lower i+I.upper i)/4
upper i := (I.lower i+3*I.upper i)/4
lower_lt_upper i := by linarith [I.lower_lt_upper i]
lemma middleBox_Icc_subset (I : Box (Fin 3)) : Box.Icc (middleBox I) ⊆ Box.Ioo I := by
intro x hx i _
have hlo := hx.1 i
have hup := hx.2 i
change (3*I.lower i+I.upper i)/4 ≤ x i at hlo
change x i ≤ (I.lower i+3*I.upper i)/4 at hup
constructor <;> linarith [I.lower_lt_upper i]
lemma middleBox_volume (I : Box (Fin 3)) :
(volume (Box.Icc (middleBox I))).toReal = (volume (Box.Ioo I)).toReal/8 := by
rw [←measure_congr (middleBox I).coe_ae_eq_Icc,
←measure_congr (I.coe_ae_eq_Icc.trans I.Ioo_ae_eq_Icc.symm)]
simp only [Box.volume_apply']
simp only [middleBox]
rw [Fin.prod_univ_three,Fin.prod_univ_three]
ring
lemma middleBox_shift_mem (I : Box (Fin 3)) {d : ℝ} (hd : 0 ≤ d)
(hsmall : ∀ i, d < (I.upper i-I.lower i)/4)
{x : Fin 3 → ℝ} (hx : x ∈ Box.Icc (middleBox I)) (j : Fin 3) :
x+Pi.single j d ∈ Box.Ioo I := by
intro i _
have hxi := middleBox_Icc_subset I hx i trivial
have hu := hx.2 i
change x i ≤ (I.lower i+3*I.upper i)/4 at hu
by_cases hij : i=j
· subst i
simp only [Pi.add_apply,Pi.single_eq_same]
constructor <;> linarith [hsmall j,hxi.1]
· simpa only [Pi.add_apply,Pi.single_eq_of_ne hij,add_zero] using hxi
lemma middleBox_eventually_shift_mem (I : Box (Fin 3)) {ε ℓ : ℝ}
(hε : 0 < ε) (hℓ : 0 < ℓ) :
∀ᶠ n : ℝ in atTop, ∀ x ∈ Box.Icc (middleBox I), ∀ j : Fin 3,
x+Pi.single j (ε/(n*ℓ)) ∈ Box.Ioo I := by
have hd : Tendsto (fun n : ℝ => ε/(n*ℓ)) atTop (𝓝 0) := by
have ht := (tendsto_const_nhds (x := ε)).mul (tendsto_inv_atTop_zero.mul_const (ℓ⁻¹))
simpa only [mul_zero,zero_mul,div_eq_mul_inv,mul_inv_rev,mul_comm (ℓ⁻¹)] using ht
have he : ∀ᶠ n : ℝ in atTop, ∀ j : Fin 3, ε/(n*ℓ) < (I.upper j-I.lower j)/4 := by
rw [Filter.eventually_all]
intro j
exact (tendsto_order.mp hd).2 _ (by linarith [I.lower_lt_upper j])
filter_upwards [he,eventually_ge_atTop (1:ℝ)] with n hn hn1
intro x hx j
exact middleBox_shift_mem I (div_nonneg hε.le (mul_nonneg (by linarith) hℓ.le)) hn hx j
end YauCounterexamples
end
end OAI
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