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import FormalConjecturesUtilBen Green's Open Problem 40
References:
[Gr24] Ben Green's Open Problem 40
[Da90] Davydov, Alexander Abramovich. "Construction of linear covering codes." Problemy Peredachi Informatsii 26.4 (1990): 38-55.
[CHL97] Cohen, G., Honkala, I., Litsyn, S., & Lobstein, A. (1997). Covering codes (Vol. 54). Elsevier.
[St94] R. Struik, Covering codes, PhD Thesis, Eindhoven University of Technology, the Netherlands, 106 pp, 1994.
open Filter Topology Fintypeopen scoped ENNReal Pointwisenamespace Green40The Hamming ball of radius $r$ in $\mathbb{F}_2^n$.
def hammingBall (n r : ℕ) : Set (𝔽₂ n) :=
{x | hammingNorm x ≤ r}$V$ is a covering subspace of $\mathbb{F}_2^n$ by $H(r)$ if $V + H(r) = \mathbb{F}_2^n$.
def IsCoveringSubspace (n r : ℕ) (V : Submodule (ZMod 2) (𝔽₂ n)) : Prop :=
(V : Set (𝔽₂ n)) + hammingBall n r = Set.univ
The minimal covering density over all covering subspaces for a given n and r.
We compute in ℝ≥0∞ (ENNReal) to gracefully handle any potential divergence.
noncomputable def minDensity (n r : ℕ) : ℝ≥0∞ :=
⨅ (V : Submodule (ZMod 2) (𝔽₂ n)) (_ : IsCoveringSubspace n r V),
(Nat.card V : ℝ≥0∞) * (Nat.card (hammingBall n r) : ℝ≥0∞) / (2 ^ n : ℝ≥0∞)Let $f(r)$ be the smallest constant such that there exists an infinite sequence of $n$'s together with subspaces $V_n \leq \mathbb{F}_2^n$ with $V_n + H(r) = \mathbb{F}_2^n$ and $|V_n| = \left(f(r) + o(1)\right) \frac{2^n}{|H(r)|}$.
noncomputable def f (r : ℕ) : ℝ≥0∞ :=
liminf (fun n ↦ minDensity n r) atTopDoes $f(r) \to \infty$? [Gr24]
@[category research open, AMS 5 94]
theorem green_40 : answer(sorry) ↔ Tendsto f atTop (𝓝 ⊤) := ⊢ True ↔ Tendsto f atTop (𝓝 ∞)
All goals completed! 🐙The only value known is $f(1) = 1$, which follows from the existence of the Hamming code [Gr24].
@[category research solved, AMS 5 94]
theorem green_40.sanity_f_one : f 1 = 1 := ⊢ f 1 = 1
All goals completed! 🐙$f(r) \le r^r / r! \sim e^r$ [Gr24].
@[category research solved, AMS 5 94]
theorem green_40.upper_bound (r : ℕ) : f r ≤ (r ^ r : ℝ≥0∞) / (r.factorial : ℝ≥0∞) := r:ℕ⊢ f r ≤ ↑r ^ r / ↑r.factorial
All goals completed! 🐙The possibility that f(r) = 1 for all r has not been ruled out [Gr24]
@[category research open, AMS 5 94]
theorem green_40.f_eq_one_for_all : answer(sorry) ↔ ∀ r, f r = 1 := ⊢ True ↔ ∀ (r : ℕ), f r = 1
All goals completed! 🐙It is not known whether f(2) = 1 [Gr24]
@[category research open, AMS 5 94]
theorem green_40.f_two_eq_one : answer(sorry) ↔ f 2 = 1 := ⊢ True ↔ f 2 = 1
All goals completed! 🐙The best-known upper bound for $f(2)$ is $1.4238$ [CHL97].
@[category research solved, AMS 5 94]
theorem green_40.upper_bound_f_two : f 2 ≤ (1.4238 : ℝ≥0∞) := ⊢ f 2 ≤ 1.4238
All goals completed! 🐙-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-- Variant with arbitrary subsets
-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
def hammingBallFinset (n r : ℕ) : Finset (𝔽₂ n) :=
Finset.univ.filter (fun x => hammingNorm x ≤ r)def IsCoveringFinset (n r : ℕ) (V : Finset (𝔽₂ n)) : Prop :=
V + hammingBallFinset n r = Finset.univnoncomputable def minDensityFinset (n r : ℕ) : ℝ≥0∞ :=
⨅ (V : Finset (𝔽₂ n)) (_ : IsCoveringFinset n r V),
(V.card : ℝ≥0∞) * (Nat.card (hammingBall n r) : ℝ≥0∞) / (2 ^ n : ℝ≥0∞)noncomputable def f_tilde (r : ℕ) : ℝ≥0∞ :=
liminf (fun n ↦ minDensityFinset n r) atTopDoes $\tilde{f}(r) \to \infty$? [Gr24]
@[category research open, AMS 5 94]
theorem green_40.variants.arbitrary_subsets : answer(sorry) ↔ Tendsto f_tilde atTop (𝓝 ⊤) := ⊢ True ↔ Tendsto f_tilde atTop (𝓝 ∞)
All goals completed! 🐙It is known that $\tilde{f}(2) = 1$ [St94].
@[category research solved, AMS 5 94]
theorem green_40.variants.arbitrary_subsets_sanity_f_tilde_two : f_tilde 2 = 1 := ⊢ f_tilde 2 = 1
All goals completed! 🐙We evidently have $\tilde{f}(r) \le f(r)$ [Gr24].
All goals completed! 🐙-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-- Variant for all n
-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
noncomputable def f_all (r : ℕ) : ℝ≥0∞ :=
limsup (fun n ↦ minDensity n r) atTopDoes $f_{\text{all}}(r) \to \infty$? [Gr24]
The target filter is 𝓝 ⊤, as in green_40 and green_40.variants.arbitrary_subsets. On
ℝ≥0∞, atTop is the principal ultrafilter at ⊤, so Tendsto f_all atTop atTop would say
that f_all r = ⊤ for all large r rather than that f_all r → ∞.
@[category research open, AMS 5 94]
theorem green_40.variants.all_n : answer(sorry) ↔ Tendsto f_all atTop (𝓝 ⊤) := ⊢ True ↔ Tendsto f_all atTop (𝓝 ∞)
All goals completed! 🐙end Green40