41.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 5 contain sorry, 0 declared as axioms

Ben Green's Open Problem 41

References

[Gr24] Ben Green's Open Problem 41
[Ma15] Manners, Freddie. "A solution to the pyjama problem." Inventiones mathematicae 202.1 (2015): 239-270.
[KrLe25] Kravitz, Noah, and James Leng. "Quantitative pyjama." arXiv preprint arXiv:2510.17744 (2025).

Imports (1)

FormalConjecturesUtil

def Green41.pyjamaSet The pyjama set is the set of points in the complex plane whose real part is within $\varepsilon$ of an integer line 40 source
Source documentation
The pyjama set is the set of points in the complex plane whose real part is within $\varepsilon$ of an integer.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set, Complex
Proof uses: Set.ofPred, Exists, Int, LE.le, abs, HSub.hSub, Complex.re
7 instances, coercions and auxiliary constants left out.
```
def pyjamaSet (ε : ℝ) : Set ℂ :=
…
```
def Green41.coveringCopies The set of valid numbers of rotated copies of the pyjama set of width ε that cover the plane line 44 source
Source documentation
The set of valid numbers of rotated copies of the pyjama set of width ε that cover the plane.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set, Nat
Proof uses: Set.ofPred, Exists, Finset, And, Eq, Finset.card, Complex, Set.iUnion, Membership.mem, HSMul.hSMul, Set.smulSet, Complex.exp, HMul.hMul, Complex.I, Green41.pyjamaSet, Set.univ
7 instances, coercions and auxiliary constants left out.
```
def coveringCopies (ε : ℝ) : Set ℕ :=
…
```
def Green41.minCopies The minimal number of rotated copies of the pyjama set of width ε needed to cover the plane line 49 source
Source documentation
The minimal number of rotated copies of the pyjama set of width ε needed to cover the plane.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: InfSet.sInf, Green41.coveringCopies
1 instances, coercions and auxiliary constants left out.
```
noncomputable def minCopies (ε : ℝ) : ℕ :=
…
```
theorem Green41.minCopies_set_nonempty [Ma15] proved that for any $\varepsilon > 0$, finitely many rotations of the pyjama set of width $\varepsilon$ cover the plane sorry line 58 source
Source documentation
[Ma15] proved that for any $\varepsilon > 0$, finitely many rotations of the pyjama set of width $\varepsilon$ cover the plane. This implies that the set we are taking the infimum over in minCopies is non-empty.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Set.Nonempty, Nat, Green41.coveringCopies
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem minCopies_set_nonempty (ε : ℝ) (hε : 0 < ε) :
    (coveringCopies ε).Nonempty := by
…
```
Mentions: Green41.coveringCopies
theorem Green41.green_41 How many rotated (about the origin) copies of the 'pyjama set' $\\{(x, y) \in \mathbb{R}^2 : \text{dist}(x, \mathbb{Z}) \leq \varepsilon\\}$ are needed to cover $\mathbb{R}^2$? sorry line 70 source
Source documentation
How many rotated (about the origin) copies of the 'pyjama set' $\\{(x, y) \in \mathbb{R}^2 : \text{dist}(x, \mathbb{Z}) \leq \varepsilon\\}$ are needed to cover $\mathbb{R}^2$?
That is, determine the minimal number of rotations as a function of $\varepsilon > 0$.

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Eq, Nat, Green41.minCopies, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_41 : ∀ ε > 0, minCopies ε = (answer(sorry) : ℝ → ℕ) ε := by
…
```
Mentions: Green41.minCopies
theorem Green41.green_41.variants.double_exponential_bound Can the triple-exponential bound of [KrLe25] be improved by one exponential? sorry line 80 source
Source documentation
Can the triple-exponential bound of [KrLe25] be improved by one exponential? That is, do $\exp\exp(\varepsilon^{-C})$ rotations suffice for some $C$ and all sufficiently small $\varepsilon > 0$? [KrLe25, Section 8.1] discusses where the three exponentials are lost and which of them one might hope to save.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Membership.mem, Set, Set.Ioc, LE.le, Green41.minCopies, Real.exp, HPow.hPow, Neg.neg
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_41.variants.double_exponential_bound : answer(sorry) ↔
    ∃ C : ℝ, ∃ ε₀ > 0, ∀ ε ∈ Ioc 0 ε₀, (minCopies ε : ℝ) ≤ Real.exp (Real.exp (ε ^ (-C))) := by
…
```
Mentions: Green41.minCopies
theorem Green41.green_41.variants.polynomial_bound Is $\varepsilon^{-C}$ rotations enough? sorry line 86 source
Source documentation
Is $\varepsilon^{-C}$ rotations enough?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Membership.mem, Set, Set.Ioc, LE.le, Green41.minCopies, HPow.hPow, Neg.neg
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_41.variants.polynomial_bound : answer(sorry) ↔
    ∃ C : ℝ, ∃ ε₀ > 0, ∀ ε ∈ Ioc 0 ε₀, (minCopies ε : ℝ) ≤ ε ^ (-C) := by
…
```
Mentions: Green41.minCopies
theorem Green41.green_41.variants.kravitz_leng [KrLe25] have established the first quantitative bound, showing via an analysis of [Ma15]'s method that $\exp\exp\exp(\varepsilon^{-C})$ rotations suffice sorry line 95 source
Source documentation
[KrLe25] have established the first quantitative bound, showing via an analysis of [Ma15]'s method that $\exp\exp\exp(\varepsilon^{-C})$ rotations suffice.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Membership.mem, Set, Set.Ioc, LE.le, Green41.minCopies, Real.exp, HPow.hPow, Neg.neg
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_41.variants.kravitz_leng :
    ∃ C : ℝ, ∃ ε₀ > 0, ∀ ε ∈ Ioc 0 ε₀, (minCopies ε : ℝ) ≤ Real.exp (Real.exp (Real.exp (ε ^ (-C)))) := by
…
```
Mentions: Green41.minCopies