openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /ShellPolynomialSubpower.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SharedBinSupportCollision | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer | |
| open Filter Asymptotics | |
| open scoped Topology | |
| theorem shell_polynomial_subpowerND (k : ℕ) (C ε : ℝ) (hε : 0<ε) : | |
| ∀ᶠ N : ℕ in atTop, C*(N:ℝ)^k ≤ ((2:ℝ)^N)^ε := by | |
| have hr : 1<(2:ℝ)^ε := Real.one_lt_rpow (by norm_num) hε | |
| have ht := (tendsto_pow_const_div_const_pow_of_one_lt k hr).const_mul C | |
| have hh : ∀ᶠ N : ℕ in atTop, C*((N:ℝ)^k/((2:ℝ)^ε)^N) ≤ 1 := by | |
| have hc := ht.eventually_lt_const (show C*0 < (1:ℝ) by simp) | |
| exact hc.mono (fun _ h => h.le) | |
| filter_upwards [hh] with N hN | |
| have he : ((2:ℝ)^ε)^N=((2:ℝ)^N)^ε := by | |
| rw [← Real.rpow_natCast ((2:ℝ)^ε) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2), | |
| ← Real.rpow_natCast (2:ℝ) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2),mul_comm ε] | |
| rw [← mul_div_assoc] at hN | |
| exact ((div_le_one (by positivity)).mp hN).trans_eq he | |
| end CurrentFalconer | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |