import Erdos1135.Tao.Probability.LogWindowBoundaryMass import Erdos1135.Tao.Probability.LogWindowEndpoints /-! # Eventual Mass Lower Bounds For Real And Floor Tao Windows This neutral proof leaf supplies the four normalizer lower bounds needed to transport the natural Proposition 1.11 rate to genuine real source windows. It uses the rounded-window mass theorem directly, without subtracting the floor-window symmetric-difference error. -/ namespace Erdos1135 namespace Tao open Filter open scoped Topology noncomputable section /-- A power-base Tao window has at least `log b / 8000` mass once its rounded width and logarithmic scale are large enough. -/ theorem log_div_eightThousand_le_taoNyOddWindow_rpow_mass {b beta : ℝ} (hb : 2 ≤ b) (hbeta : 1 ≤ beta) (hlog : 8000 ≤ Real.log b) (hwidth : b ^ beta + 2 ≤ (b ^ beta) ^ taoAlpha) : Real.log b / 8000 ≤ logFinsetMass (taoNyOddWindow (b ^ beta) taoAlpha) := by have hbpos : 0 < b := by linarith have hbone : 1 ≤ b := by linarith have hbetaNonneg : 0 ≤ beta := by linarith have hyTwo : 2 ≤ b ^ beta := by exact hb.trans (Real.self_le_rpow_of_one_le hbone hbeta) have hyNonneg : 0 ≤ b ^ beta := (Real.rpow_pos_of_pos hbpos beta).le have hab : Nat.ceil (b ^ beta) ≤ Nat.floor ((b ^ beta) ^ taoAlpha) := by have hsucc := taoNyLo_succ_le_taoNyHi_of_width hyNonneg hwidth simpa only [taoNyLo, taoNyHi] using (show taoNyLo (b ^ beta) ≤ taoNyHi (b ^ beta) taoAlpha by omega) have hlogY : Real.log (b ^ beta) = beta * Real.log b := Real.log_rpow hbpos beta have hlogBNonneg : 0 ≤ Real.log b := by linarith have hlogScale : Real.log b ≤ beta * Real.log b := by simpa only [one_mul] using mul_le_mul_of_nonneg_right hbeta hlogBNonneg have hlogWidth : 8 ≤ (taoAlpha - 1) * Real.log (b ^ beta) := by rw [hlogY] norm_num [taoAlpha] nlinarith have hmass := one_eighth_log_width_le_rounded_power_oddWindow_mass hyTwo taoAlpha_one_lt hab hlogWidth calc Real.log b / 8000 ≤ Real.log (b ^ beta) / 8000 := by rw [hlogY] exact div_le_div_of_nonneg_right hlogScale (by norm_num) _ = ((taoAlpha - 1) / 8) * Real.log (b ^ beta) := by norm_num [taoAlpha] ring _ ≤ logFinsetMass (taoNyOddWindow (b ^ beta) taoAlpha) := by simpa only [taoNyOddWindow, taoNyLo, taoNyHi] using hmass /-- Every fixed power scale `beta >= 1` eventually has the standard Tao window mass lower bound. -/ theorem eventually_log_div_eightThousand_le_taoNyOddWindow_rpow_mass {beta : ℝ} (hbeta : 1 ≤ beta) : ∀ᶠ b : ℝ in atTop, Real.log b / 8000 ≤ logFinsetMass (taoNyOddWindow (b ^ beta) taoAlpha) := by have hbetaPos : 0 < beta := zero_lt_one.trans_le hbeta have hpow : Tendsto (fun b : ℝ => b ^ beta) atTop atTop := tendsto_rpow_atTop hbetaPos have hlog : Tendsto (fun b : ℝ => Real.log b) atTop atTop := Real.tendsto_log_atTop filter_upwards [eventually_ge_atTop (2 : ℝ), hlog.eventually_ge_atTop (8000 : ℝ), hpow.eventually taoNy_width_eventually] with b hb hlogLarge hwidth exact log_div_eightThousand_le_taoNyOddWindow_rpow_mass hb hbeta hlogLarge hwidth /-- For `B = floor x`, both the floor-based and genuine real power windows eventually have mass at least `log B / 8000`. -/ theorem eventually_floor_real_rpow_window_mass_lower {beta : ℝ} (hbeta : 1 ≤ beta) : ∀ᶠ x : ℝ in atTop, Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ beta) taoAlpha) ∧ Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (x ^ beta) taoAlpha) := by have hfloorCast : Tendsto (fun x : ℝ => ((Nat.floor x : ℕ) : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop.comp tendsto_nat_floor_atTop have hfloorMass := hfloorCast.eventually (eventually_log_div_eightThousand_le_taoNyOddWindow_rpow_mass hbeta) have hrealMass := eventually_log_div_eightThousand_le_taoNyOddWindow_rpow_mass hbeta filter_upwards [hfloorMass, hrealMass, eventually_ge_atTop (2 : ℝ)] with x hfloorMass hrealMass hx refine ⟨hfloorMass, ?_⟩ have hxpos : 0 < x := by linarith have hx0 : 0 ≤ x := hxpos.le have hfloorOne : 1 ≤ Nat.floor x := Nat.le_floor (show ((1 : ℕ) : ℝ) ≤ x by norm_num; linarith) have hfloorPos : 0 < ((Nat.floor x : ℕ) : ℝ) := by exact_mod_cast (show 0 < Nat.floor x by omega) have hlogLe : Real.log ((Nat.floor x : ℕ) : ℝ) ≤ Real.log x := Real.strictMonoOn_log.monotoneOn hfloorPos hxpos (Nat.floor_le hx0) exact (div_le_div_of_nonneg_right hlogLe (by norm_num)).trans hrealMass /-- Eventual normalizer packet for the two Proposition 1.11 source scales. -/ structure TaoProp111RealFloorWindowMassFacts (x : ℝ) : Prop where one_le_x : 1 ≤ x log_floor_large : 8000 ≤ Real.log ((Nat.floor x : ℕ) : ℝ) alpha_floor : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ taoAlpha) taoAlpha) alpha_real : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (x ^ taoAlpha) taoAlpha) alphaSq_floor : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (((Nat.floor x : ℕ) : ℝ) ^ (taoAlpha ^ 2)) taoAlpha) alphaSq_real : Real.log ((Nat.floor x : ℕ) : ℝ) / 8000 ≤ logFinsetMass (taoNyOddWindow (x ^ (taoAlpha ^ 2)) taoAlpha) theorem TaoProp111RealFloorWindowMassFacts.log_floor_pos {x : ℝ} (facts : TaoProp111RealFloorWindowMassFacts x) : 0 < Real.log ((Nat.floor x : ℕ) : ℝ) := by linarith [facts.log_floor_large] /-- The four real/floor Proposition 1.11 source normalizers eventually satisfy one common `log(floor x) / 8000` lower bound. -/ theorem eventually_taoProp111RealFloorWindowMassFacts : ∀ᶠ x : ℝ in atTop, TaoProp111RealFloorWindowMassFacts x := by have hfloorCast : Tendsto (fun x : ℝ => ((Nat.floor x : ℕ) : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop.comp tendsto_nat_floor_atTop have hlogFloor : Tendsto (fun x : ℝ => Real.log ((Nat.floor x : ℕ) : ℝ)) atTop atTop := Real.tendsto_log_atTop.comp hfloorCast filter_upwards [eventually_ge_atTop (1 : ℝ), hlogFloor.eventually_ge_atTop (8000 : ℝ), eventually_floor_real_rpow_window_mass_lower (show (1 : ℝ) ≤ taoAlpha by norm_num [taoAlpha]), eventually_floor_real_rpow_window_mass_lower (show (1 : ℝ) ≤ taoAlpha ^ 2 by norm_num [taoAlpha])] with x hx hlog hAlpha hAlphaSq exact ⟨hx, hlog, hAlpha.1, hAlpha.2, hAlphaSq.1, hAlphaSq.2⟩ end end Tao end Erdos1135