| import OAI.MeasureTheory.Falconer.Fourier.SymbolProductBounds | |
| import Mathlib.Analysis.SpecialFunctions.SmoothTransition | |
| import Mathlib.Algebra.Order.Floor.Ring | |
| namespace OAI | |
| namespace PlanarFalconer | |
| open Set | |
| open scoped ContDiff Topology | |
| noncomputable def angularBinKernel (t : ℝ) : ℝ := | |
| Real.smoothTransition (t+1) - Real.smoothTransition t | |
| theorem angularBinKernel_contDiff : ContDiff ℝ ∞ angularBinKernel := | |
| (Real.smoothTransition.contDiff.comp (contDiff_id.add contDiff_const)).sub | |
| Real.smoothTransition.contDiff | |
| theorem angularBinKernel_nonneg (t : ℝ) : 0 ≤ angularBinKernel t := | |
| sub_nonneg.mpr (Real.smoothTransition.monotone (by linarith)) | |
| theorem angularBinKernel_le_one (t : ℝ) : angularBinKernel t ≤ 1 := | |
| sub_le_iff_le_add.mpr (by linarith [Real.smoothTransition.le_one (t+1), | |
| Real.smoothTransition.nonneg t]) | |
| theorem angularBinKernel_zero_left {t : ℝ} (ht : t ≤ -1) : angularBinKernel t = 0 := by | |
| unfold angularBinKernel | |
| rw [Real.smoothTransition.zero_of_nonpos (by linarith), | |
| Real.smoothTransition.zero_of_nonpos (by linarith), sub_self] | |
| theorem angularBinKernel_zero_right {t : ℝ} (ht : 1 ≤ t) : angularBinKernel t = 0 := by | |
| unfold angularBinKernel | |
| rw [Real.smoothTransition.one_of_one_le (by linarith), | |
| Real.smoothTransition.one_of_one_le ht, sub_self] | |
| theorem angularBinKernel_tsupport : tsupport angularBinKernel ⊆ Icc (-1 : ℝ) 1 := by | |
| apply closure_minimal _ isClosed_Icc | |
| intro t ht | |
| constructor | |
| · by_contra h | |
| exact ht (angularBinKernel_zero_left (le_of_not_ge h)) | |
| · by_contra h | |
| exact ht (angularBinKernel_zero_right (le_of_not_ge h)) | |
| theorem angularBinKernel_compact : HasCompactSupport angularBinKernel := | |
| isCompact_Icc.of_isClosed_subset isClosed_closure angularBinKernel_tsupport | |
| theorem angularBinKernel_two_terms {t : ℝ} (h0 : 0 ≤ t) (h1 : t ≤ 1) : | |
| angularBinKernel t + angularBinKernel (t-1) = 1 := by | |
| unfold angularBinKernel | |
| rw [Real.smoothTransition.one_of_one_le (show 1 ≤ t+1 by linarith), | |
| Real.smoothTransition.zero_of_nonpos (show t-1 ≤ 0 by linarith), | |
| sub_add_cancel] | |
| ring | |
| theorem angularBinKernel_translate_zero (t : ℝ) (k : ℤ) | |
| (hk : k ≠ ⌊t⌋ ∧ k ≠ ⌊t⌋+1) : angularBinKernel (t-k) = 0 := by | |
| have hlow := Int.floor_le t | |
| have hhigh := Int.lt_floor_add_one t | |
| by_cases h : k ≤ ⌊t⌋-1 | |
| · apply angularBinKernel_zero_right | |
| have hh : (k : ℝ) ≤ (⌊t⌋ : ℝ)-1 := by exact_mod_cast h | |
| linarith | |
| · apply angularBinKernel_zero_left | |
| have hh : ⌊t⌋+2 ≤ k := by omega | |
| have hh : (⌊t⌋ : ℝ)+2 ≤ (k : ℝ) := by exact_mod_cast hh | |
| linarith | |
| theorem angularBinKernel_floor_partition (t : ℝ) : | |
| angularBinKernel (t-(⌊t⌋ : ℤ)) + angularBinKernel (t-(⌊t⌋+1 : ℤ)) = 1 := by | |
| have h := angularBinKernel_two_terms (sub_nonneg.mpr (Int.floor_le t)) | |
| (show t-(⌊t⌋ : ℤ) ≤ 1 by linarith [Int.lt_floor_add_one t]) | |
| simpa only [Int.cast_add, Int.cast_one, sub_add_eq_sub_sub] using h | |
| end PlanarFalconer | |
| end OAI | |