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Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Sphere Packing inter ball encard ge

SpherePacking.inter_ball_encard_ge

sphere packingFourier analysismodular forms

Plain-language statement

This gives an upper bound on the number of points in the sphere packing X with norm less than R.

Exact Lean statement

theorem SpherePacking.inter_ball_encard_ge (hd : 0 < d) (R : ℝ) :
    (S.centers ∩ ball 0 R).encard ≥
      volume (S.balls ∩ ball 0 (R - S.separation / 2))
        / volume (ball (0 : EuclideanSpace ℝ (Fin d)) (S.separation / 2))

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem SpherePacking.inter_ball_encard_ge (hd : 0 < d) (R : ℝ) :    (S.centers ∩ ball 0 R).encard ≥      volume (S.balls ∩ ball 0 (R - S.separation / 2))        / volume (ball (0 : EuclideanSpace ℝ (Fin d)) (S.separation / 2)) := by  have h := volume.mono <|    biUnion_balls_inter_subset_biUnion_inter_balls S.centers (S.separation / 2) R  change volume _ ≤ volume _ at h  simp_rw [Set.biUnion_eq_iUnion, S.volume_iUnion_balls_eq_tsum _ (le_refl _),    Measure.addHaar_ball_center, ENNReal.tsum_set_const] at h  haveI : Nonempty (Fin d) := Fin.pos_iff_nonempty.mp hd  rwa [← ENNReal.div_le_iff_le_mul] at h <;> left  · exact (volume_ball_pos _ (by linarith [S.separation_pos])).ne.symm  · exact (volume_ball_lt_top _).ne
Project
Sphere Packing in Dimension 8
License
Apache-2.0
Commit
acfc6204e65a
Source
SpherePacking/Basic/SpherePacking.lean:411-423

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Related declarations

Project-declaredLean 4.31.0

Anti Der Pos

antiDerPos

Plain-language statement

If FF is a modular form where F(it)F(it) is positive for sufficiently large tt (i.e. constant term is positive) and the derivative is positive, then FF is also positive.

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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Project-declaredLean 4.31.0

Anti Serre Der Pos

antiSerreDerPos

Plain-language statement

Let F:H→CF : \mathbb{H} \to \mathbb{C} be a holomorphic function where F(it)F(it) is real for all t>0t > 0. Assume that Serre derivative ∂kF\partial_k F is positive on the imaginary axis. If F(it)F(it) is positive for sufficiently large tt, then F(it)F(it) is positive for all t>0t > 0.

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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