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b224933 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | 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
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