| import OAI.NumberTheory.TwoPointCorrelations.HalaszSmoothBounds |
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| namespace OAI |
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| /-! A fixed-power change of the prime cutoff changes the squared |
| pretentious distance by an absolute constant. -/ |
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| namespace TwoPointCorrelations |
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| open Finset Complex |
| open scoped ComplexConjugate |
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| lemma halasz_prime_cutoff_subset {n N : ℕ} (hn : n ≤ N) : primesUpTo n ⊆ primesUpTo N := by |
| intro p hp |
| obtain ⟨hp, hprime⟩ := mem_filter.mp hp |
| exact mem_filter.mpr ⟨mem_range.mpr (by have := mem_range.mp hp; omega), hprime⟩ |
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| theorem halasz_distance_cutoff_loss : ∃ K : ℝ, 0 ≤ K ∧ |
| ∀ (f : ℕ → ℂ), OneBounded f → ∀ (n N : ℕ), 2 ≤ n → n ≤ N → N ≤ n ^ 3 → |
| ∀ t : ℝ, squaredDistance f (mrtArchimedeanTwist t) N ≤ |
| squaredDistance f (mrtArchimedeanTwist t) n + K := by |
| obtain ⟨C, hC⟩ := primeReciprocalInput |
| let K := 2 * (Real.log 3 + 2 * |C|) |
| have hK : 0 ≤ K := by dsimp [K]; positivity |
| refine ⟨K, hK, ?_⟩ |
| intro f hf n N hn hnN hNcube t |
| have hn0 : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n) |
| have hN0 : (0 : ℝ) < N := by exact_mod_cast (by omega : 0 < N) |
| have hln : 0 < Real.log (n : ℝ) := Real.log_pos (by exact_mod_cast (by omega : 1 < n)) |
| have hlN : 0 < Real.log (N : ℝ) := Real.log_pos (by exact_mod_cast (by omega : 1 < N)) |
| have hlNn : Real.log (N : ℝ) ≤ 3 * Real.log (n : ℝ) := by |
| have h := Real.log_le_log hN0 (show (N : ℝ) ≤ (n : ℝ) ^ 3 by exact_mod_cast hNcube) |
| simpa only [Real.log_pow, Nat.cast_ofNat] using h |
| have hll : Real.log (Real.log (N : ℝ)) - Real.log (Real.log (n : ℝ)) ≤ Real.log 3 := by |
| have h := Real.log_le_log hlN hlNn |
| rw [Real.log_mul (by norm_num) hln.ne'] at h |
| linarith |
| have hmN := hC (N : ℝ) (by exact_mod_cast (show 2 ≤ N by omega)) |
| have hmn := hC (n : ℝ) (by exact_mod_cast hn) |
| rw [mrt_sievePrimesUpTo_nat] at hmN hmn |
| have hmass : (∑ p ∈ primesUpTo N, (1 : ℝ) / p) - |
| (∑ p ∈ primesUpTo n, (1 : ℝ) / p) ≤ Real.log 3 + 2 * |C| := by |
| have ha := (abs_le.mp hmN).2 |
| have hb := (abs_le.mp hmn).1 |
| linarith [le_abs_self C] |
| let P := primesUpTo N \ primesUpTo n |
| have hsub := halasz_prime_cutoff_subset hnN |
| have hterm (p : ℕ) (hp : p ∈ P) : |
| (1 - (f p * conj (mrtArchimedeanTwist t p)).re) / (p : ℝ) ≤ 2 * (1 / (p : ℝ)) := by |
| have hprime : p.Prime := (mem_filter.mp (mem_sdiff.mp hp).1).2 |
| have hnrm : ‖f p * conj (mrtArchimedeanTwist t p)‖ ≤ 1 := by |
| simpa only [norm_mul, norm_conj, mrtArchimedeanTwist_norm, mul_one] using hf p hprime.pos |
| have hre := (neg_le_abs _).trans ((Complex.abs_re_le_norm _).trans hnrm) |
| have hnum : 1 - (f p * conj (mrtArchimedeanTwist t p)).re ≤ 2 := by linarith |
| simpa only [mul_one_div] using div_le_div_of_nonneg_right hnum (Nat.cast_nonneg p) |
| have hs := sum_le_sum (fun p hp => hterm p hp) |
| rw [← mul_sum] at hs |
| have heDist := sum_sdiff hsub (f := fun p : ℕ => |
| (1 - (f p * conj (mrtArchimedeanTwist t p)).re) / (p : ℝ)) |
| have heMass := sum_sdiff hsub (f := fun p : ℕ => (1 : ℝ) / p) |
| change (∑ p ∈ P, (1 - (f p * conj (mrtArchimedeanTwist t p)).re) / (p : ℝ)) + |
| squaredDistance f (mrtArchimedeanTwist t) n = squaredDistance f (mrtArchimedeanTwist t) N at heDist |
| change (∑ p ∈ P, (1 : ℝ) / p) + (∑ p ∈ primesUpTo n, (1 : ℝ) / p) = |
| (∑ p ∈ primesUpTo N, (1 : ℝ) / p) at heMass |
| dsimp [K] |
| linarith |
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| end TwoPointCorrelations |
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| end OAI |
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