24.lean`

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

7 declarations, 4 contain sorry, 0 declared as axioms

Green's Open Problem 24

References:

Green, Ben. "100 open problems." (2024).
[Aa19] Aaronson, James. "Maximising the number of solutions to a linear equation in a set of integers." Bulletin of the London Mathematical Society 51.4 (2019): 577-594.
[HaL28] Hardy, G. H., and J. E. Littlewood. "Notes on the theory of series (VIII): an inequality." Journal of the London Mathematical Society 1.2 (1928): 105-110.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def Green24.max013AffineTranslates The maximum number of $\lbrace 0,1,3 \rbrace$ affine translates that a set of size $n$ can contain line 41 source
Source documentation
The maximum number of $\lbrace 0,1,3 \rbrace$ affine translates that a set of size $n$ can contain.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, Int, And, Eq, Finset.card, Prod, Finset.filter, Green24.max013AffineTranslates.match_1, Ne, Membership.mem, HAdd.hAdd, HMul.hMul, OfNat.ofNat, HSub.hSub, Prod.fst, Prod.snd, SProd.sprod
15 instances, coercions and auxiliary constants left out.
```
noncomputable def max013AffineTranslates (n : ℕ) : ℕ :=
…
```
theorem Green24.green_24 If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\lbrace 0,1,3 \rbrace$ that $A$ can contain? sorry line 56 source
Source documentation
If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\lbrace 0,1,3 \rbrace$ that $A$ can contain?
Conjectured in [Aa19] p.579: $\left(\frac{1}{3} + o(1)\right) n^2$.

depends on sorry what this means
Statement uses: Nat, Eq, Green24.max013AffineTranslates, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_24 : ∀ n, max013AffineTranslates n = (answer(sorry) : ℕ → ℕ) n := by
…
```
Mentions: Green24.max013AffineTranslates
theorem Green24.variants.upper_trivial From [Aa19] p.577: the trivial upper bound is $n^2$ (non asymptotic) line 66 source
Source documentation
From [Aa19] p.577: the trivial upper bound is $n^2$ (non asymptotic).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, Green24.max013AffineTranslates, HPow.hPow, OfNat.ofNat
Proof uses: csSup_le, Set.ofPred, Exists, Finset, Int, And, Eq, Finset.card, Prod, Finset.filter, Green24.max013AffineTranslates.match_1, Ne, Membership.mem, HAdd.hAdd, HMul.hMul, HSub.hSub, Prod.fst, Prod.snd, SProd.sprod, Exists.intro and 19 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem upper_trivial {n : ℕ} : max013AffineTranslates n ≤ n ^ 2 := by
…
```
Mentions: Green24.max013AffineTranslates
def Green24.variants.gamma The asymptotic constant $\gamma$ defined in [Aa19] p.579 line 75 source
Source documentation
The asymptotic constant $\gamma$ defined in [Aa19] p.579.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Filter.limsup, Nat, HDiv.hDiv, Green24.max013AffineTranslates, HPow.hPow, OfNat.ofNat, Filter.atTop
13 instances, coercions and auxiliary constants left out.
```
noncomputable def gamma : ℝ :=
…
```
theorem Green24.variants.upper_HL Asymptotic upper bound (1.2) in [Aa19] sorry line 80 source
Source documentation
Asymptotic upper bound (1.2) in [Aa19]. Named after Hardy and Littlewood [HaL28].

depends on sorry what this means
Statement uses: LE.le, Real, Green24.variants.gamma, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem upper_HL : gamma ≤ 3/4 := by
…
```
Mentions: Green24.variants.gamma
theorem Green24.variants.lower_HL Asymptotic lower bound (1.2) in [Aa19] sorry line 85 source
Source documentation
Asymptotic lower bound (1.2) in [Aa19]. Named after Hardy and Littlewood [HaL28].

depends on sorry what this means
Statement uses: GE.ge, Real, Green24.variants.gamma, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem lower_HL : gamma ≥ 1/12 := by
…
```
Mentions: Green24.variants.gamma
theorem Green24.variants.conjecture Conjecture p.579 in [Aa19]: $\left(\frac{1}{3} + o(1)\right) n^2$ sorry line 90 source
Source documentation
Conjecture p.579 in [Aa19]: $\left(\frac{1}{3} + o(1)\right) n^2$.

depends on sorry what this means
Statement uses: Eq, Real, Green24.variants.gamma, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture : gamma = 1/3 := by
…
```
Mentions: Green24.variants.gamma