51.lean`

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

8 declarations, 6 contain sorry, 0 declared as axioms

Green's Open Problem 51

References:

[Gr24] Green's Open Problems
[Gr13] B. J. Green, Restriction and Kakeya phenomena, notes from a 2003 course. Available at http://people.maths.ox.ac.uk/greenbj/papers/rkp.pdf
[Sa11] Sanders, Tom. "Green's sumset problem at density one half." Acta Arithmetica 146.1 (2011): 91-101.
[Gr02] Green, Ben. "Arithmetic progressions in sumsets." Geometric & Functional Analysis GAFA 12.3 (2002): 584-597.
[Ruz91] Ruzsa, Imre Z. "Arithmetic progressions in sumsets." Acta Arithmetica 60.2 (1991): 191-202.

Imports (1)

FormalConjecturesUtil

def Green51.guaranteedMaxCosetDim The largest dimension of a coset guaranteed to be contained in $2A$ for $A \subseteq \mathbb{F}_2^n$ with density $\alpha$ line 43 source
Source documentation
The largest dimension of a coset guaranteed to be contained in $2A$ for $A \subseteq \mathbb{F}_2^n$ with density $\alpha$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, 𝔽₂, GE.ge, Finset.dens, Fin, ZMod, OfNat.ofNat, Eq, maxCosetDim, HAdd.hAdd
34 instances, coercions and auxiliary constants left out.
```
noncomputable def guaranteedMaxCosetDim (n : ℕ) (α : ℝ) : ℕ :=
…
```
theorem Green51.green_51 Suppose that $A \subset \mathbb{F}_2^n$ is a set of density $\alpha$ sorry line 54 source
Source documentation
Suppose that $A \subset \mathbb{F}_2^n$ is a set of density $\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?
We phrase this by asking for the exact function $F(\alpha, n)$ giving the maximum dimension of a guaranteed coset.

depends on sorry what this means
Statement uses: Eq, Nat, Real, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Green51.guaranteedMaxCosetDim
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_51 : answer(sorry) = guaranteedMaxCosetDim := by
…
```
Mentions: Green51.guaranteedMaxCosetDim
theorem Green51.green_51.lower It is known that $A + A$ must contain a coset of dimension $\gg_\alpha n$ [Gr13] sorry line 59 source
Source documentation
It is known that $A + A$ must contain a coset of dimension $\gg_\alpha n$ [Gr13].

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, LE.le, Exists, And, GT.gt, Filter.Eventually, Nat, HMul.hMul, Green51.guaranteedMaxCosetDim, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_51.lower :
    ∀ (α : ℝ), 0 < α → α ≤ 1 →
    ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, c * (n : ℝ) ≤ guaranteedMaxCosetDim n α := by
…
```
Mentions: c, Green51.guaranteedMaxCosetDim
theorem Green51.green_51.upper It is known that $A + A$ need not contain a coset of dimension $n - \sqrt{n}$ [Gr13] sorry line 66 source
Source documentation
It is known that $A + A$ need not contain a coset of dimension $n - \sqrt{n}$ [Gr13].

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, LE.le, Filter.Eventually, Nat, LT.lt, Green51.guaranteedMaxCosetDim, HSub.hSub, Real.sqrt, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_51.upper :
    ∃ α > 0, α ≤ 1 ∧ ∀ᶠ (n : ℕ) in atTop, (guaranteedMaxCosetDim n α : ℝ) < (n : ℝ) - sqrt n := by
…
```
Mentions: Green51.guaranteedMaxCosetDim
theorem Green51.green_51.one_half Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha > 1/2 - C/\sqrt{n}$ sorry line 75 source
Source documentation
Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha > 1/2 - C/\sqrt{n}$. Does $A + A$ contain a subspace of co-dimension $O_C(1)$? [Sa11, Question 5.1]

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, Nat, Filter.Eventually, GT.gt, HSub.hSub, HDiv.hDiv, Real.sqrt, LE.le, HAdd.hAdd, Green51.guaranteedMaxCosetDim, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_51.one_half :
    answer(sorry) ↔ ∀ (k : ℝ), 0 < k →
      ∃ (c : ℕ), ∀ᶠ (n : ℕ) in atTop,
        ∀ (α : ℝ), α > (1/2 : ℝ) - k / sqrt (n : ℝ) → α ≤ 1 →
          n ≤ guaranteedMaxCosetDim n α + c := by
…
```
Mentions: c, Green51.guaranteedMaxCosetDim
def Green51.guaranteedMaxAPLength The largest length of an arithmetic progression guaranteed to be contained in $A+A$ for $A \subseteq \{1, \dots, N\}$ with density $\alpha$ line 86 source
Source documentation
The largest length of an arithmetic progression guaranteed to be contained in $A+A$ for $A \subseteq \{1, \dots, N\}$ with density $\alpha$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, LE.le, Finset.Icc, OfNat.ofNat, HMul.hMul, Finset.card, Eq, SupSet.sSup, Set, And, HAdd.hAdd, Set.IsAPOfLength, ENat
21 instances, coercions and auxiliary constants left out.
```
noncomputable def guaranteedMaxAPLength (N : ℕ) (α : ℝ) : ℕ :=
…
```
theorem Green51.green_51.lower_ap It is known that $A + A$ must contain an arithmetic progression of length $\sim \exp(c (\log N)^{1/2})$ [Gr02] sorry line 92 source
Source documentation
It is known that $A + A$ must contain an arithmetic progression of length $\sim \exp(c (\log N)^{1/2})$ [Gr02].

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, LE.le, Exists, And, GT.gt, Filter.Eventually, Nat, Real.exp, HMul.hMul, HPow.hPow, Real.log, HDiv.hDiv, Green51.guaranteedMaxAPLength, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_51.lower_ap :
    ∀ (α : ℝ), 0 < α → α ≤ 1 →
      ∃ c > 0, ∀ᶠ (N : ℕ) in atTop,
        exp (c * log (N : ℝ) ^ (1/2 : ℝ)) ≤ guaranteedMaxAPLength N α := by
…
```
Mentions: c, Green51.guaranteedMaxAPLength
theorem Green51.green_51.upper_ap It is known that $A + A$ need not contain an arithmetic progression of length $\sim \exp(c (\log N)^{2/3 + \varepsilon})$, for any $\varepsilon > 0$ [Ruz91] sorry line 103 source
Source documentation
It is known that $A + A$ need not contain an arithmetic progression of length $\sim \exp(c (\log N)^{2/3 + \varepsilon})$, for any $\varepsilon > 0$ [Ruz91].

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, HDiv.hDiv, Nat, Exists, And, GT.gt, Filter.Eventually, LE.le, Green51.guaranteedMaxAPLength, Real.exp, HMul.hMul, HPow.hPow, Real.log, HAdd.hAdd, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_51.upper_ap :
    ∀ (α : ℝ), 0 < α → α < 1/2 → ∀ (ε : ℝ), 0 < ε →
      ∃ c > 0, ∀ᶠ (N : ℕ) in atTop,
        guaranteedMaxAPLength N α ≤ exp (c * log (N : ℝ) ^ (2/3 + ε : ℝ)) := by
…
```
Mentions: c, Green51.guaranteedMaxAPLength