import Erdos1135.Tao.Probability.LogSupportPerturbation import Erdos1135.Tao.Probability.LogWindowEndpoints import Erdos1135.Tao.Probability.ReciprocalProgression /-! # Floor Perturbations Of Tao Logarithmic Windows This neutral proof leaf compares two ordered odd logarithmic windows. It keeps the first reciprocal endpoint in the shell estimate, so later real-to-natural window adapters can retain a polynomial error rate. -/ namespace Erdos1135 namespace Tao open scoped BigOperators noncomputable section /-- On positive naturals, the shifted project-local logarithmic weight is the ordinary reciprocal weight. -/ theorem logNatWeight_eq_one_div_of_pos {n : ℕ} (hn : 0 < n) : logNatWeight n = 1 / (n : ℝ) := by rcases n with _ | n · omega simp [logNatWeight, logWeight] /-- Dropping the oddness filter can only increase logarithmic mass. -/ theorem logFinsetMass_oddLogWindow_le_sum_Icc_reciprocal {A U : ℕ} (hA : 1 ≤ A) : logFinsetMass (oddLogWindow A U) ≤ ∑ N ∈ Finset.Icc A U, 1 / (N : ℝ) := by classical unfold logFinsetMass calc (∑ N ∈ oddLogWindow A U, logNatWeight N) = ∑ N ∈ oddLogWindow A U, 1 / (N : ℝ) := by apply Finset.sum_congr rfl intro N hN rw [logNatWeight_eq_one_div_of_pos] have hAN := (oddLogWindow_mem.mp hN).1 omega _ ≤ ∑ N ∈ Finset.Icc A U, 1 / (N : ℝ) := by apply Finset.sum_le_sum_of_subset_of_nonneg · intro N hN exact Finset.mem_Icc.mpr ⟨(oddLogWindow_mem.mp hN).1, (oddLogWindow_mem.mp hN).2.1⟩ · intro N _hN _hnot positivity /-- For ordered endpoints, values lost from the first window lie in its lower endpoint shell. -/ theorem oddLogWindow_sdiff_subset_lowerShell {lo₀ hi₀ lo₁ hi₁ : ℕ} (hhi : hi₀ ≤ hi₁) : oddLogWindow lo₀ hi₀ \ oddLogWindow lo₁ hi₁ ⊆ oddLogWindow lo₀ (lo₁ - 1) := by intro N hN rw [Finset.mem_sdiff] at hN rw [oddLogWindow_mem] at hN ⊢ rcases hN with ⟨⟨hlo₀N, hNhi₀, hodd⟩, hnot⟩ have hnot' : ¬ (lo₁ ≤ N ∧ N ≤ hi₁ ∧ N % 2 = 1) := by simpa only [oddLogWindow_mem] using hnot exact ⟨hlo₀N, by omega, hodd⟩ /-- For ordered endpoints, values gained by the second window lie in its upper endpoint shell. -/ theorem oddLogWindow_sdiff_subset_upperShell {lo₀ hi₀ lo₁ hi₁ : ℕ} (hlo : lo₀ ≤ lo₁) : oddLogWindow lo₁ hi₁ \ oddLogWindow lo₀ hi₀ ⊆ oddLogWindow (hi₀ + 1) hi₁ := by intro N hN rw [Finset.mem_sdiff] at hN rw [oddLogWindow_mem] at hN ⊢ rcases hN with ⟨⟨hlo₁N, hNhi₁, hodd⟩, hnot⟩ have hnot' : ¬ (lo₀ ≤ N ∧ N ≤ hi₀ ∧ N % 2 = 1) := by simpa only [oddLogWindow_mem] using hnot exact ⟨by omega, hNhi₁, hodd⟩ /-- Ordered endpoint movement is controlled by the two endpoint shells. -/ theorem logFinsetSymmDiffMass_oddLogWindow_le_shells {lo₀ hi₀ lo₁ hi₁ : ℕ} (hlo : lo₀ ≤ lo₁) (hhi : hi₀ ≤ hi₁) : logFinsetSymmDiffMass (oddLogWindow lo₀ hi₀) (oddLogWindow lo₁ hi₁) ≤ logFinsetMass (oddLogWindow lo₀ (lo₁ - 1)) + logFinsetMass (oddLogWindow (hi₀ + 1) hi₁) := by unfold logFinsetSymmDiffMass exact add_le_add (logFinsetMass_mono (oddLogWindow_sdiff_subset_lowerShell hhi)) (logFinsetMass_mono (oddLogWindow_sdiff_subset_upperShell hlo)) /-- A reciprocal endpoint shell produced by moving a base from `B` to a real `x < B + 1` has mass at most `3 / B`, uniformly for exponents in `[1, 2]`. The assumptions on `A` and `U` deliberately expose the two rounding facts needed by lower and upper endpoint shells. -/ theorem logFinsetMass_oddLogWindow_le_three_div_of_floor_shell {B A U : ℕ} {x gamma : ℝ} (hB : 1 ≤ B) (hBx : (B : ℝ) ≤ x) (hxB : x ≤ (B : ℝ) + 1) (hgammaOne : 1 ≤ gamma) (hgammaTwo : gamma ≤ 2) (hBA : (B : ℝ) ^ gamma ≤ (A : ℝ)) (hUx : (U : ℝ) ≤ x ^ gamma) : logFinsetMass (oddLogWindow A U) ≤ 3 / (B : ℝ) := by have hBpos : 0 < (B : ℝ) := by exact_mod_cast (show 0 < B by omega) have hBone : (1 : ℝ) ≤ (B : ℝ) := by exact_mod_cast hB have hxpos : 0 < x := hBpos.trans_le hBx have hgammaNonneg : 0 ≤ gamma := by linarith have hBself : (B : ℝ) ≤ (B : ℝ) ^ gamma := Real.self_le_rpow_of_one_le hBone hgammaOne have hBAcast : (B : ℝ) ≤ (A : ℝ) := hBself.trans hBA have hBA_nat : B ≤ A := by exact_mod_cast hBAcast have hA : 1 ≤ A := hB.trans hBA_nat by_cases hAU : A ≤ U · have hApos : 0 < (A : ℝ) := by exact_mod_cast (show 0 < A by omega) have hUpos : 0 < (U : ℝ) := by exact hApos.trans_le (by exact_mod_cast hAU) have hBpowPos : 0 < (B : ℝ) ^ gamma := Real.rpow_pos_of_pos hBpos gamma have hxpowPos : 0 < x ^ gamma := Real.rpow_pos_of_pos hxpos gamma have hlogU : Real.log (U : ℝ) ≤ Real.log (x ^ gamma) := Real.strictMonoOn_log.monotoneOn hUpos hxpowPos hUx have hlogB : Real.log ((B : ℝ) ^ gamma) ≤ Real.log (A : ℝ) := Real.strictMonoOn_log.monotoneOn hBpowPos hApos hBA have hlogGap : Real.log x - Real.log (B : ℝ) ≤ 1 / (B : ℝ) := by have hratioPos : 0 < x / (B : ℝ) := div_pos hxpos hBpos have hlogRatio := Real.log_le_sub_one_of_pos hratioPos have hratio : x / (B : ℝ) - 1 ≤ 1 / (B : ℝ) := by apply (sub_le_iff_le_add).2 calc x / (B : ℝ) ≤ ((B : ℝ) + 1) / (B : ℝ) := by exact div_le_div_of_nonneg_right hxB hBpos.le _ = 1 / (B : ℝ) + 1 := by field_simp ring rw [Real.log_div hxpos.ne' hBpos.ne'] at hlogRatio exact hlogRatio.trans hratio have hgammaLog : gamma * (Real.log x - Real.log (B : ℝ)) ≤ 2 / (B : ℝ) := by calc gamma * (Real.log x - Real.log (B : ℝ)) ≤ gamma * (1 / (B : ℝ)) := mul_le_mul_of_nonneg_left hlogGap hgammaNonneg _ ≤ 2 * (1 / (B : ℝ)) := by exact mul_le_mul_of_nonneg_right hgammaTwo (one_div_nonneg.mpr hBpos.le) _ = 2 / (B : ℝ) := by ring have hlogShell : Real.log ((U : ℝ) / (A : ℝ)) ≤ 2 / (B : ℝ) := by rw [Real.log_rpow hxpos] at hlogU rw [Real.log_rpow hBpos] at hlogB rw [Real.log_div hUpos.ne' hApos.ne'] linarith have hfirst : 1 / (A : ℝ) ≤ 1 / (B : ℝ) := one_div_le_one_div_of_le hBpos hBAcast calc logFinsetMass (oddLogWindow A U) ≤ ∑ N ∈ Finset.Icc A U, 1 / (N : ℝ) := logFinsetMass_oddLogWindow_le_sum_Icc_reciprocal hA _ ≤ 1 / (A : ℝ) + Real.log ((U : ℝ) / (A : ℝ)) := sum_Icc_reciprocal_le_one_div_add_log_div hA hAU _ ≤ 1 / (B : ℝ) + 2 / (B : ℝ) := add_le_add hfirst hlogShell _ = 3 / (B : ℝ) := by ring · have hUA : U < A := by omega have hempty : oddLogWindow A U = ∅ := by simp [oddLogWindow, Finset.Icc_eq_empty_of_lt hUA] rw [hempty] simpa [logFinsetMass] using (div_nonneg (show (0 : ℝ) ≤ 3 by norm_num) hBpos.le) private theorem natCast_ceil_sub_one_le {z : ℝ} (hz : 0 < z) : ((Nat.ceil z - 1 : ℕ) : ℝ) ≤ z := by have hceil : (Nat.ceil z : ℝ) < z + 1 := Nat.ceil_lt_add_one hz.le have hceilPos : 1 ≤ Nat.ceil z := by have : (0 : ℝ) < Nat.ceil z := lt_of_lt_of_le hz (Nat.le_ceil z) exact_mod_cast this push_cast [Nat.cast_sub hceilPos] linarith private theorem le_natCast_floor_add_one (z : ℝ) : z ≤ ((Nat.floor z + 1 : ℕ) : ℝ) := by push_cast exact (Nat.lt_floor_add_one z).le /-- Rounding the two endpoints of an ordered real-power window changes its logarithmic source mass by at most `6 / B`. The two `3 / B` contributions are the lower and upper endpoint shells. -/ theorem logFinsetSymmDiffMass_taoNyOddWindow_rpow_le_six_div {B : ℕ} {x beta alpha : ℝ} (hB : 1 ≤ B) (hBx : (B : ℝ) ≤ x) (hxB : x ≤ (B : ℝ) + 1) (hbetaOne : 1 ≤ beta) (hbetaTwo : beta ≤ 2) (halphaNonneg : 0 ≤ alpha) (hprodOne : 1 ≤ beta * alpha) (hprodTwo : beta * alpha ≤ 2) : logFinsetSymmDiffMass (taoNyOddWindow ((B : ℝ) ^ beta) alpha) (taoNyOddWindow (x ^ beta) alpha) ≤ 6 / (B : ℝ) := by have hBpos : 0 < (B : ℝ) := by exact_mod_cast (show 0 < B by omega) have hxpos : 0 < x := hBpos.trans_le hBx have hbetaNonneg : 0 ≤ beta := by linarith have hBpowPos : 0 < (B : ℝ) ^ beta := Real.rpow_pos_of_pos hBpos beta have hxpowPos : 0 < x ^ beta := Real.rpow_pos_of_pos hxpos beta have hinner : (B : ℝ) ^ beta ≤ x ^ beta := Real.rpow_le_rpow hBpos.le hBx hbetaNonneg have houter : ((B : ℝ) ^ beta) ^ alpha ≤ (x ^ beta) ^ alpha := Real.rpow_le_rpow hBpowPos.le hinner halphaNonneg have hlo : Nat.ceil ((B : ℝ) ^ beta) ≤ Nat.ceil (x ^ beta) := Nat.ceil_mono hinner have hhi : Nat.floor (((B : ℝ) ^ beta) ^ alpha) ≤ Nat.floor ((x ^ beta) ^ alpha) := Nat.floor_mono houter have hlower : logFinsetMass (oddLogWindow (Nat.ceil ((B : ℝ) ^ beta)) (Nat.ceil (x ^ beta) - 1)) ≤ 3 / (B : ℝ) := by apply logFinsetMass_oddLogWindow_le_three_div_of_floor_shell hB hBx hxB hbetaOne hbetaTwo · exact Nat.le_ceil ((B : ℝ) ^ beta) · exact natCast_ceil_sub_one_le hxpowPos have hupper : logFinsetMass (oddLogWindow (Nat.floor (((B : ℝ) ^ beta) ^ alpha) + 1) (Nat.floor ((x ^ beta) ^ alpha))) ≤ 3 / (B : ℝ) := by apply logFinsetMass_oddLogWindow_le_three_div_of_floor_shell hB hBx hxB hprodOne hprodTwo · rw [← Real.rpow_mul hBpos.le] exact le_natCast_floor_add_one ((B : ℝ) ^ (beta * alpha)) · rw [← Real.rpow_mul hxpos.le] exact Nat.floor_le (Real.rpow_nonneg hxpos.le (beta * alpha)) have hshells := logFinsetSymmDiffMass_oddLogWindow_le_shells hlo hhi change logFinsetSymmDiffMass (oddLogWindow (Nat.ceil ((B : ℝ) ^ beta)) (Nat.floor (((B : ℝ) ^ beta) ^ alpha))) (oddLogWindow (Nat.ceil (x ^ beta)) (Nat.floor ((x ^ beta) ^ alpha))) ≤ 6 / (B : ℝ) calc logFinsetSymmDiffMass (oddLogWindow (Nat.ceil ((B : ℝ) ^ beta)) (Nat.floor (((B : ℝ) ^ beta) ^ alpha))) (oddLogWindow (Nat.ceil (x ^ beta)) (Nat.floor ((x ^ beta) ^ alpha))) ≤ logFinsetMass (oddLogWindow (Nat.ceil ((B : ℝ) ^ beta)) (Nat.ceil (x ^ beta) - 1)) + logFinsetMass (oddLogWindow (Nat.floor (((B : ℝ) ^ beta) ^ alpha) + 1) (Nat.floor ((x ^ beta) ^ alpha))) := hshells _ ≤ 3 / (B : ℝ) + 3 / (B : ℝ) := add_le_add hlower hupper _ = 6 / (B : ℝ) := by ring /-- The ordered-window estimate specialized to `B = floor x`. -/ theorem logFinsetSymmDiffMass_taoNyOddWindow_floor_rpow_le_six_div {x beta alpha : ℝ} (hx : 1 ≤ x) (hbetaOne : 1 ≤ beta) (hbetaTwo : beta ≤ 2) (halphaNonneg : 0 ≤ alpha) (hprodOne : 1 ≤ beta * alpha) (hprodTwo : beta * alpha ≤ 2) : logFinsetSymmDiffMass (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ beta) alpha) (taoNyOddWindow (x ^ beta) alpha) ≤ 6 / ((Nat.floor x : ℕ) : ℝ) := by have hx0 : 0 ≤ x := zero_le_one.trans hx have hB : 1 ≤ Nat.floor x := Nat.le_floor (show ((1 : ℕ) : ℝ) ≤ x by simpa using hx) apply logFinsetSymmDiffMass_taoNyOddWindow_rpow_le_six_div hB (Nat.floor_le hx0) (Nat.lt_floor_add_one x).le hbetaOne hbetaTwo halphaNonneg hprodOne hprodTwo /-- The first Proposition 1.11 source scale, `beta = alpha`, has the sharp `6 / floor x` weighted support perturbation. -/ theorem logFinsetSymmDiffMass_taoNyOddWindow_floor_taoAlpha_le_six_div {x : ℝ} (hx : 1 ≤ x) : logFinsetSymmDiffMass (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ taoAlpha) taoAlpha) (taoNyOddWindow (x ^ taoAlpha) taoAlpha) ≤ 6 / ((Nat.floor x : ℕ) : ℝ) := by apply logFinsetSymmDiffMass_taoNyOddWindow_floor_rpow_le_six_div hx all_goals norm_num [taoAlpha] /-- The second Proposition 1.11 source scale, `beta = alpha^2`, obeys the same uniform weighted support perturbation. -/ theorem logFinsetSymmDiffMass_taoNyOddWindow_floor_taoAlpha_sq_le_six_div {x : ℝ} (hx : 1 ≤ x) : logFinsetSymmDiffMass (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ (taoAlpha ^ 2)) taoAlpha) (taoNyOddWindow (x ^ (taoAlpha ^ 2)) taoAlpha) ≤ 6 / ((Nat.floor x : ℕ) : ℝ) := by apply logFinsetSymmDiffMass_taoNyOddWindow_floor_rpow_le_six_div hx all_goals norm_num [taoAlpha] /-- A `6 / B` weighted support error and the standard `log B / 8000` normalizer lower bound give the sharp uniform event error `96000 / (B * log B)`. -/ theorem abs_logFinsetProb_sub_le_ninetySixThousand_div_mul_log {B : ℕ} {S T : Finset ℕ} (hB : 1 ≤ B) (hlog : 0 < Real.log (B : ℝ)) (hsymm : logFinsetSymmDiffMass S T ≤ 6 / (B : ℝ)) (hmassS : Real.log (B : ℝ) / 8000 ≤ logFinsetMass S) (hmassT : Real.log (B : ℝ) / 8000 ≤ logFinsetMass T) (E : Set ℕ) : |logFinsetProb S E - logFinsetProb T E| ≤ 96000 / ((B : ℝ) * Real.log (B : ℝ)) := by have hBpos : 0 < (B : ℝ) := by exact_mod_cast (show 0 < B by omega) have hbasePos : 0 < Real.log (B : ℝ) / 8000 := div_pos hlog (by norm_num) have hSpos : 0 < logFinsetMass S := hbasePos.trans_le hmassS have hTpos : 0 < logFinsetMass T := hbasePos.trans_le hmassT have hmin : Real.log (B : ℝ) / 8000 ≤ min (logFinsetMass S) (logFinsetMass T) := le_min hmassS hmassT have hminPos : 0 < min (logFinsetMass S) (logFinsetMass T) := hbasePos.trans_le hmin have hnum : 2 * logFinsetSymmDiffMass S T ≤ 12 / (B : ℝ) := by calc 2 * logFinsetSymmDiffMass S T ≤ 2 * (6 / (B : ℝ)) := mul_le_mul_of_nonneg_left hsymm (by norm_num) _ = 12 / (B : ℝ) := by ring have hnumNonneg : 0 ≤ 12 / (B : ℝ) := div_nonneg (by norm_num) hBpos.le calc |logFinsetProb S E - logFinsetProb T E| ≤ 2 * logFinsetSymmDiffMass S T / min (logFinsetMass S) (logFinsetMass T) := abs_logFinsetProb_sub_le_two_symmDiffMass_div_min hSpos hTpos E _ ≤ (12 / (B : ℝ)) / min (logFinsetMass S) (logFinsetMass T) := div_le_div_of_nonneg_right hnum hminPos.le _ ≤ (12 / (B : ℝ)) / (Real.log (B : ℝ) / 8000) := div_le_div_of_nonneg_left hnumNonneg hbasePos hmin _ = 96000 / ((B : ℝ) * Real.log (B : ℝ)) := by field_simp [hBpos.ne', hlog.ne'] norm_num /-- Normalized real-to-floor source perturbation for a general power scale whose two endpoint exponents lie in `[1, 2]`. -/ theorem abs_logFinsetProb_taoNyOddWindow_floor_rpow_sub_le {x beta alpha : ℝ} (hx : 1 ≤ x) (hbetaOne : 1 ≤ beta) (hbetaTwo : beta ≤ 2) (halphaNonneg : 0 ≤ alpha) (hprodOne : 1 ≤ beta * alpha) (hprodTwo : beta * alpha ≤ 2) (hlog : 0 < Real.log ((Nat.floor x : ℕ) : ℝ)) (hmassFloor : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ beta) alpha)) (hmassReal : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (x ^ beta) alpha)) (E : Set ℕ) : |logFinsetProb (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ beta) alpha) E - logFinsetProb (taoNyOddWindow (x ^ beta) alpha) E| ≤ 96000 / (((Nat.floor x : ℕ) : ℝ) * Real.log ((Nat.floor x : ℕ) : ℝ)) := by have hB : 1 ≤ Nat.floor x := Nat.le_floor (show ((1 : ℕ) : ℝ) ≤ x by simpa using hx) exact abs_logFinsetProb_sub_le_ninetySixThousand_div_mul_log hB hlog (logFinsetSymmDiffMass_taoNyOddWindow_floor_rpow_le_six_div hx hbetaOne hbetaTwo halphaNonneg hprodOne hprodTwo) hmassFloor hmassReal E /-- Normalized source perturbation at the first Proposition 1.11 scale. -/ theorem abs_logFinsetProb_taoNyOddWindow_floor_taoAlpha_sub_le {x : ℝ} (hx : 1 ≤ x) (hlog : 0 < Real.log ((Nat.floor x : ℕ) : ℝ)) (hmassFloor : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ taoAlpha) taoAlpha)) (hmassReal : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (x ^ taoAlpha) taoAlpha)) (E : Set ℕ) : |logFinsetProb (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ taoAlpha) taoAlpha) E - logFinsetProb (taoNyOddWindow (x ^ taoAlpha) taoAlpha) E| ≤ 96000 / (((Nat.floor x : ℕ) : ℝ) * Real.log ((Nat.floor x : ℕ) : ℝ)) := by apply abs_logFinsetProb_taoNyOddWindow_floor_rpow_sub_le hx (beta := taoAlpha) (alpha := taoAlpha) · norm_num [taoAlpha] · norm_num [taoAlpha] · norm_num [taoAlpha] · norm_num [taoAlpha] · norm_num [taoAlpha] · exact hlog · exact hmassFloor · exact hmassReal /-- Normalized source perturbation at the second Proposition 1.11 scale. -/ theorem abs_logFinsetProb_taoNyOddWindow_floor_taoAlpha_sq_sub_le {x : ℝ} (hx : 1 ≤ x) (hlog : 0 < Real.log ((Nat.floor x : ℕ) : ℝ)) (hmassFloor : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ (taoAlpha ^ 2)) taoAlpha)) (hmassReal : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (x ^ (taoAlpha ^ 2)) taoAlpha)) (E : Set ℕ) : |logFinsetProb (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ (taoAlpha ^ 2)) taoAlpha) E - logFinsetProb (taoNyOddWindow (x ^ (taoAlpha ^ 2)) taoAlpha) E| ≤ 96000 / (((Nat.floor x : ℕ) : ℝ) * Real.log ((Nat.floor x : ℕ) : ℝ)) := by apply abs_logFinsetProb_taoNyOddWindow_floor_rpow_sub_le hx (beta := taoAlpha ^ 2) (alpha := taoAlpha) · norm_num [taoAlpha] · norm_num [taoAlpha] · norm_num [taoAlpha] · norm_num [taoAlpha] · norm_num [taoAlpha] · exact hlog · exact hmassFloor · exact hmassReal end end Tao end Erdos1135