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import FormalConjecturesUtilGreen's Open Problem 42
[Gr24] Green, Ben. "100 open problems." (2024).
[CoEl03] Cohn, Henry, and Noam Elkies. "New upper bounds on sphere packings I." Annals of Mathematics (2003): 689-714.
[Vi17] Viazovska, Maryna S. "The sphere packing problem in dimension 8." Annals of mathematics (2017): 991-1015.
[CKM17] Cohn, H., Kumar, A., Miller, S., Radchenko, D., & Viazovska, M. (2017). The sphere packing problem in dimension 24. Annals of mathematics, 185(3), 1017-1033.
[Sa21] Sardari, Naser Talebizadeh. "Higher Fourier interpolation on the plane." arXiv preprint arXiv:2102.08753 (2021).
open Real Complex MeasureTheoryopen scoped EuclideanGeometry FourierTransform
namespace Green42
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasureSpace V] [BorelSpace V] [FiniteDimensional ℝ V]
The real-valued Fourier transform used in the Cohn--Elkies conditions.
For real radial admissible functions, the complex Fourier transform is expected
to be real-valued; we take .re to expose the real scalar used in the inequality.
Convention: Mathlib's 𝓕 f w expands to $\int e^{-2\pi i \langle v, w \rangle} f(v) dv$, which
matches [CoEl03]'s $\hat{f}(t) = \int f(x) e^{-2\pi i \langle x, t \rangle} dx$.
noncomputable def fHat (f : V → ℝ) (t : V) : ℝ :=
(𝓕 (fun x ↦ (f x : ℂ)) t).re
Definition 2.1 from [CoEl03]: A function is admissible if both the function and its Fourier transform decay sufficiently fast.
def CohnElkiesAdmissible (f : V → ℝ) : Prop :=
∃ C > 0, ∃ δ > 0,
(∀ x : V, |f x| ≤ C / (1 + ‖x‖) ^ ((Module.finrank ℝ V : ℝ) + δ)) ∧
(∀ t : V, |fHat f t| ≤ C / (1 + ‖t‖) ^ ((Module.finrank ℝ V : ℝ) + δ))
The structural rules a function must satisfy to successfully pass through
the Cohn-Elkies scheme and generate
def SatisfiesCohnElkiesScheme (f : V → ℝ) : Prop :=
CohnElkiesAdmissible f ∧
(∀ x y : V, ‖x‖ = ‖y‖ → f x = f y) ∧ -- Radial symmetry
(∀ x : V, 2 ≤ ‖x‖ → f x ≤ 0) ∧ -- Spatial constraint (minimum distance 2)
(∀ t : V, 0 ≤ fHat f t) ∧ -- Frequency positivity
(0 < fHat f 0) ∧ -- Non-zero frequency at the origin
(0 < f 0) -- Positive value at the origi
The statement that there exists a function in dimension d satisfying the Cohn-Elkies
scheme which achieves the center density bound bound.
def CohnElkiesOptimal (d : ℕ) (bound : ℝ) : Prop :=
∃ f : ℝ^d → ℝ,
SatisfiesCohnElkiesScheme f ∧
f 0 / fHat f 0 = bound
Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?
@[category research open, AMS 51 52]
theorem green_42 :
answer(sorry) ↔ CohnElkiesOptimal 2 (Real.sqrt 3 / 6) := ⊢ True ↔ CohnElkiesOptimal 2 (√3 / 6)
All goals completed! 🐙[CoEl03] proved this when $d = 1$.
@[category research solved, AMS 51 52]
theorem green_42.variants.dimension_one :
CohnElkiesOptimal 1 (1 / 2) := ⊢ CohnElkiesOptimal 1 (1 / 2)
All goals completed! 🐙[Vi17] established the case $d = 8$.
@[category research solved, AMS 51 52, formal_proof using lean4 at "https://github.com/math-inc/Sphere-Packing-Lean/blob/main/SpherePacking/Dim8/MainTheorem.lean"]
theorem green_42.variants.dimension_eight :
CohnElkiesOptimal 8 (1 / 16) := ⊢ CohnElkiesOptimal 8 (1 / 16)
All goals completed! 🐙In [CKM17], [Vi17] was adapted to $d = 24$.
@[category research solved, AMS 51 52, formal_proof using lean4 at "https://github.com/math-inc/Sphere-Packing-Lean/blob/main/SpherePacking/Dim24/MainTheorem.lean"]
theorem green_42.variants.dimension_twenty_four :
CohnElkiesOptimal 24 1 := ⊢ CohnElkiesOptimal 24 1
All goals completed! 🐙
end Green42