| import OAI.NumberTheory.Jacobsthal.Estimates.EvenThresholdVolume |
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| namespace OAI |
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| namespace Erdos970 |
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| section |
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| namespace ErdosEvenThreshold |
| open Set MeasureTheory NumberTheoryLean |
| open FinitePathGeometry DerivativeWeights TransitionKernels FinitePathMeasures KernelDensityBridge |
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| noncomputable def stripDensityBound : ℝ := (5/4:ℝ)*phiEven 2/phiOdd 5 |
| noncomputable def stripConstant : ℝ := 20*stripDensityBound |
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| theorem stripDensityBound_pos : 0 < stripDensityBound := by |
| have he := phiEven_pos (by norm_num : (1:ℝ)<2) |
| have ho := phiOdd_pos 5 |
| unfold stripDensityBound |
| positivity |
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| theorem stripConstant_pos : 0 < stripConstant := by |
| unfold stripConstant |
| exact mul_pos (by norm_num) stripDensityBound_pos |
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| theorem oddDensity_compact_bound (s : OddState) {t : ℝ} (ht2 : 2 ≤ t) (ht4 : t ≤ 4) : |
| oddDensity s t ≤ stripDensityBound := by |
| change (if max 2 (s.1-1) ≤ t then W t*phiEven t/phiOdd s.1 else 0) ≤ _ |
| by_cases hsupp : max 2 (s.1-1) ≤ t |
| · rw [ite_eq_left hsupp] |
| have hs5 : s.1 ≤ 5 := by linarith [le_max_right (2:ℝ) (s.1-1)] |
| have ht0 : 0 < t := by linarith |
| have hW : W t ≤ (5:ℝ)/4 := by |
| unfold W |
| apply (div_le_div_iff₀ (sq_pos_of_pos ht0) (by norm_num : (0:ℝ)<4)).mpr |
| nlinarith |
| have hphi : phiEven t ≤ phiEven 2 := |
| phiEven_strictAntiOn.antitoneOn (by norm_num) (by change 1 < t; linarith : t ∈ Ioi (1:ℝ)) ht2 |
| have hden : phiOdd 5 ≤ phiOdd s.1 := phiOdd_antitone hs5 |
| have hepos : 0 < phiEven 2 := phiEven_pos (by norm_num) |
| have htpos : 0 < phiEven t := phiEven_pos (by linarith) |
| have hnum := mul_le_mul hW hphi htpos.le (by norm_num : (0:ℝ)≤5/4) |
| calc |
| _ ≤ ((5:ℝ)/4*phiEven 2)/phiOdd s.1 := div_le_div_of_nonneg_right hnum (phiOdd_pos _).le |
| _ ≤ ((5:ℝ)/4*phiEven 2)/phiOdd 5 := |
| div_le_div_of_nonneg_left (by positivity) (phiOdd_pos 5) hden |
| _ = _ := rfl |
| · rw [ite_eq_right hsupp] |
| exact stripDensityBound_pos.le |
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|
| theorem oddKernel_strip_bound (s : OddState) {R d : ℝ} |
| (hR : 0 < R) (hd0 : 0 ≤ d) (hd1 : d ≤ 1) : |
| oddKernel s (thresholdStrip R d) ≤ ENNReal.ofReal (stripConstant*d/R) := by |
| rw [oddKernel_apply] |
| calc |
| _ ≤ ∫⁻ _t in thresholdStrip R d,ENNReal.ofReal stripDensityBound := by |
| apply setLIntegral_mono measurable_const |
| intro t ht |
| have h4 := (strip_parameters hd1 ht).2 |
| exact ENNReal.ofReal_le_ofReal (oddDensity_compact_bound s ht.1 h4) |
| _ = ENNReal.ofReal stripDensityBound*volume (thresholdStrip R d) := by simp |
| _ ≤ ENNReal.ofReal stripDensityBound*ENNReal.ofReal (20*d/R) := |
| mul_le_mul_right (thresholdStrip_volume hR hd0 hd1) _ |
| _ = ENNReal.ofReal (stripConstant*d/R) := by |
| rw [← ENNReal.ofReal_mul stripDensityBound_pos.le] |
| congr 1 |
| unfold stripConstant |
| ring |
|
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| theorem oddToEven_strip_bound (s : OddState) {R d : ℝ} |
| (hR : 0 < R) (hd0 : 0 ≤ d) (hd1 : d ≤ 1) : |
| oddToEven s ((Subtype.val : EvenState → ℝ) ⁻¹' thresholdStrip R d) ≤ |
| ENNReal.ofReal (stripConstant*d/R) := by |
| have hh := oddKernel_strip_bound s hR hd0 hd1 |
| rw [← oddToEven_map_ratio s,Measure.map_apply measurable_subtype_coe (thresholdStrip_measurable R d)] at hh |
| exact hh |
| |
| theorem exists_uniform_strip_bound : ∃ C : ℝ,0 < C ∧ ∀ R d : ℝ,0 < R → 0 ≤ d → d ≤ 1 → |
| ∀ s : OddState,oddToEven s ((Subtype.val : EvenState → ℝ) ⁻¹' thresholdStrip R d) ≤ |
| ENNReal.ofReal (C*d/R) := |
| ⟨stripConstant,stripConstant_pos,fun _ _ hR hd0 hd1 s => oddToEven_strip_bound s hR hd0 hd1⟩ |
|
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| end ErdosEvenThreshold |
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| end |
|
|
| section |
|
|
| namespace ErdosEvenThreshold |
| open Set MeasureTheory NumberTheoryLean |
| open FinitePathGeometry TransitionKernels FinitePathMeasures |
|
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| noncomputable def nearThreshold (R d : ℝ) : Set ℝ := |
| {t | 1 < nextExponent R t ∧ 0 ≤ thresholdSlack R t ∧ thresholdSlack R t ≤ d} |
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| theorem thresholdStrip_eq_near (R d : ℝ) : thresholdStrip R d=nearThreshold R d := by |
| ext t |
| constructor |
| · intro ht |
| exact ht.2 |
| · rintro ⟨hx,hlo,hhi⟩ |
| have hm := gap_eq_exponent_mul R t |
| have hlow := le_max_left (2*nextExponent R t) (nextExponent R t+2) |
| have ht2 : 2 ≤ t := by |
| have hh : nextExponent R t*2 ≤ nextExponent R t*t := by |
| unfold thresholdSlack at hlo |
| nlinarith |
| exact le_of_mul_le_mul_left hh (by linarith : 0 < nextExponent R t) |
| exact ⟨ht2,hx,hlo,hhi⟩ |
|
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| theorem nearThreshold_measurable (R d : ℝ) : MeasurableSet (nearThreshold R d) := by |
| rw [← thresholdStrip_eq_near] |
| exact thresholdStrip_measurable R d |
|
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| theorem conditional_even_near_threshold (s : OddState) {R d : ℝ} |
| (hR : 0 < R) (hd0 : 0 ≤ d) (hd1 : d ≤ 1) : |
| oddToEven s ((Subtype.val : EvenState → ℝ) ⁻¹' nearThreshold R d) ≤ |
| ENNReal.ofReal (stripConstant*d/R) := by |
| rw [← thresholdStrip_eq_near] |
| exact oddToEven_strip_bound s hR hd0 hd1 |
| |
| theorem conditional_even_near_threshold_real (s : OddState) {R d : ℝ} |
| (hR : 0 < R) (hd0 : 0 ≤ d) (hd1 : d ≤ 1) : |
| (oddToEven s ((Subtype.val : EvenState → ℝ) ⁻¹' nearThreshold R d)).toReal ≤ stripConstant*d/R := by |
| have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top (conditional_even_near_threshold s hR hd0 hd1) |
| have hnon : 0 ≤ stripConstant*d/R := div_nonneg (mul_nonneg stripConstant_pos.le hd0) hR.le |
| simpa only [ENNReal.toReal_ofReal hnon] using hh |
|
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| theorem exists_conditional_even_bound : ∃ C : ℝ,0 < C ∧ ∀ R d : ℝ,0 < R → 0 ≤ d → d ≤ 1 → |
| ∀ s : OddState,(oddToEven s ((Subtype.val : EvenState → ℝ) ⁻¹' nearThreshold R d)).toReal ≤ C*d/R := |
| ⟨stripConstant,stripConstant_pos,fun _ _ hR hd0 hd1 s => conditional_even_near_threshold_real s hR hd0 hd1⟩ |
| |
| end ErdosEvenThreshold |
| |
| end |
| |
| end Erdos970 |
| |
| end OAI |
| |
