| import OAI.NumberTheory.TwoPoint.Halasz.HalaszCollisionMoment |
| import OAI.NumberTheory.TwoPoint.Halasz.HalaszVinogradovProduct |
|
|
| namespace OAI |
|
|
| /-! Absorbing the contribution of repeated long coordinates by the |
| diagonal lower bound for the complete mean value. -/ |
| namespace TwoPointCorrelations |
|
|
| lemma halasz_collision_algebra {k : ℕ} (hk : 0<k) {A B I C R : ℝ} |
| (hA : 0<A) (hB : 0<B) (hC : 0≤C) (hR : C^2<R) |
| (hdiag : R^k*B≤A) (hmoment : I^(2*k)≤A^(2*k-1)*B) : C*I<A := by |
| by_contra hn |
| have hAI : A≤C*I := le_of_not_gt hn |
| have hpow := pow_le_pow_left₀ hA.le hAI (2*k) |
| rw [mul_pow] at hpow |
| have ha : A^(2*k) ≤ C^(2*k)*A^(2*k-1)*B := by |
| calc |
| A^(2*k) ≤ C^(2*k)*I^(2*k) := hpow |
| _ ≤ C^(2*k)*(A^(2*k-1)*B) := |
| mul_le_mul_of_nonneg_left hmoment (pow_nonneg hC _) |
| _ = _ := by ring |
| have hb := halasz_collision_power_absorb (by omega : 1≤2*k) hA ha |
| have hRpow : C^(2*k)<R^k := by |
| rw [pow_mul] |
| exact pow_lt_pow_left₀ hR (sq_nonneg C) (by omega) |
| have hstrict : C^(2*k)*B<R^k*B := mul_lt_mul_of_pos_right hRpow hB |
| linarith |
|
|
| theorem halasz_collision_moment_small {s k N : ℕ} (hs : 0<s) (hk : 2≤k) |
| {C : ℝ} (hC : 0≤C) (hN : C^2<N) : |
| C*halaszCollisionMoment s k N < (halaszVinogradovCount (s+k) k N:ℝ) := by |
| have hN0 : 0<N := by exact_mod_cast lt_of_le_of_lt (sq_nonneg C) hN |
| have hJs : 0<(halaszVinogradovCount s k N:ℝ) := by |
| exact_mod_cast lt_of_lt_of_le (Nat.pow_pos hN0) (halasz_vinogradov_count_lower s k N) |
| have hJ : 0<(halaszVinogradovCount (s+k) k N:ℝ) := by |
| exact_mod_cast lt_of_lt_of_le (Nat.pow_pos hN0) |
| (halasz_vinogradov_count_lower (s+k) k N) |
| have hdiag : (N:ℝ)^k*(halaszVinogradovCount s k N:ℝ) ≤ |
| (halaszVinogradovCount (s+k) k N:ℝ) := by |
| exact_mod_cast (by simpa only [Nat.add_comm k s] using |
| halasz_vinogradov_diagonal_extension s k N) |
| exact halasz_collision_algebra (by omega : 0<k) hJ hJs hC hN hdiag |
| (halasz_collision_moment_bound hs hk N) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
