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d4317d7 | 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 | import OAI.MeasureTheory.Falconer.Recursion.ProfileEndpointStability
import Mathlib.Algebra.Order.Floor.Ring
namespace OAI
/-! Inward rounding of profile intervals to a fixed finite scale grid. -/
namespace Falconer
noncomputable def profileGridLeft (n : ℕ) (a : ℝ) : ℝ := (⌈(n : ℝ)*a⌉ : ℝ)/(n : ℝ)
noncomputable def profileGridRight (n : ℕ) (b : ℝ) : ℝ := (⌊(n : ℝ)*b⌋ : ℝ)/(n : ℝ)
theorem profileGridLeft_bounds {n : ℕ} (hn : 0 < n) (a : ℝ) :
a ≤ profileGridLeft n a ∧ profileGridLeft n a ≤ a+1/(n : ℝ) := by
have hnR : (0 : ℝ) < n := by exact_mod_cast hn
constructor
· apply (le_div_iff₀ hnR).2
simpa only [mul_comm] using Int.le_ceil ((n : ℝ)*a)
· apply (div_le_iff₀ hnR).2
have hb := (Int.ceil_lt_add_one ((n : ℝ)*a)).le
have he : (a+1/(n : ℝ))*(n : ℝ) = (n : ℝ)*a+1 := by field_simp
rw [he]
exact hb
theorem profileGridRight_bounds {n : ℕ} (hn : 0 < n) (b : ℝ) :
b-1/(n : ℝ) ≤ profileGridRight n b ∧ profileGridRight n b ≤ b := by
have hnR : (0 : ℝ) < n := by exact_mod_cast hn
constructor
· apply (le_div_iff₀ hnR).2
have hb := (Int.lt_floor_add_one ((n : ℝ)*b)).le
have he : (b-1/(n : ℝ))*(n : ℝ) = (n : ℝ)*b-1 := by field_simp
rw [he]
linarith
· apply (div_le_iff₀ hnR).2
simpa only [mul_comm] using Int.floor_le ((n : ℝ)*b)
theorem profileGrid_interval_bounds {n : ℕ} (hn : 0 < n) (a h : ℝ) :
profileGridRight n (a+h)-profileGridLeft n a ≤ h ∧
h-2/(n : ℝ) ≤ profileGridRight n (a+h)-profileGridLeft n a ∧
|profileGridLeft n a-a| ≤ 1/(n : ℝ) ∧
|profileGridRight n (a+h)-(a+h)| ≤ 1/(n : ℝ) := by
have hl := profileGridLeft_bounds hn a
have hr := profileGridRight_bounds hn (a+h)
have he : 2/(n : ℝ) = 2*(1/(n : ℝ)) := by ring
refine ⟨by linarith,by rw [he]; linarith,?_,?_⟩
· rw [abs_of_nonneg (sub_nonneg.mpr hl.1)]
linarith
· rw [abs_of_nonpos (sub_nonpos.mpr hr.2)]
linarith
end Falconer
end OAI
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