54.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Ben Green's Open Problem 54

References:

Ben Green's Open Problem 54
Original formulation: M. Talagrand, Are All Sets of Positive Measure Essentially Convex?, in Operator Theory: Advances and Applications, 77, 1995 Birkhäuser Verlag Basel/Switzerland.
Hua–Song–Tudose, On Talagrand's convexity conjecture (2026)
M. Talagrand, Some open problems, §2

Here $nK = K + \cdots + K$ ($n$ times) is the $n$-fold sumset of $K$, written n • K for n : ℕ with open scoped Pointwise.

Imports (1)

FormalConjecturesUtil

def Green54.gaussianMeasureInf The infinite-dimensional Gaussian measure γ∞ on ℝ^ℕ, defined as the countable product of standard Gaussian measures line 44 source
Source documentation
The infinite-dimensional Gaussian measure γ∞ on ℝ^ℕ, defined as the countable product of standard Gaussian measures.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasureTheory.Measure, Nat, Real
Proof uses: MeasureTheory.Measure.infinitePi, ProbabilityTheory.gaussianReal, OfNat.ofNat, NNReal
7 instances, coercions and auxiliary constants left out.
```
noncomputable def gaussianMeasureInf : Measure (ℕ → ℝ) :=
…
```
theorem Green54.green_54 Documented declaration: green_54 sorry line 57 source
Source documentation
Let $K \subset \mathbb{R}^{\mathbb{N}}$ be a balanced compact set (that is, $\lambda K \subseteq K$ whenever $|\lambda| \leq 1$) and suppose that the normalised Gaussian measure $\gamma_\infty(K) \geq 0.99$. Does the sumset $10K = K + \cdots + K$ ($10$ times) contain a compact convex set $C$ with $\gamma_\infty(C) \geq 0.01$?
The answer is yes: Hua, Song and Tudose proved that if $\gamma_n(A) > 5/6$ then $3(A + A + A)$ contains a symmetric convex body $C$ with $\gamma_n(C) \geq 1/4$, uniformly in $n$.

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Real, IsCompact, Balanced, LE.le, ENNReal, OfScientific.ofScientific, Bool.true, MeasureTheory.Measure, Green54.gaussianMeasureInf, Exists, And, Convex, HSMul.hSMul, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
58 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_54 :
    answer(True) ↔ ∀ K : Set (ℕ → ℝ), IsCompact K → Balanced ℝ K → (0.99 : ℝ≥0∞) ≤
    gaussianMeasureInf K → ∃ C : Set (ℕ → ℝ), IsCompact C ∧ Convex ℝ C ∧ C ⊆ (10 : ℕ) • K ∧
    (0.01 : ℝ≥0∞) ≤ gaussianMeasureInf C := by
…
```
Mentions: Green54.gaussianMeasureInf
theorem Green54.green_54_known_case The same statement is known to be false for the sumset $2K = K + K$ instead of $10K$ sorry line 67 source
Source documentation
The same statement is known to be false for the sumset $2K = K + K$ instead of $10K$.

depends on sorry what this means
Statement uses: Not, Set, Nat, Real, IsCompact, Balanced, LE.le, ENNReal, OfScientific.ofScientific, Bool.true, MeasureTheory.Measure, Green54.gaussianMeasureInf, Exists, And, Convex, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
53 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_54_known_case : ¬ (∀ K : Set (ℕ → ℝ), IsCompact K → Balanced ℝ K → (0.99 : ℝ≥0∞) ≤
    gaussianMeasureInf K → ∃ C : Set (ℕ → ℝ), IsCompact C ∧ Convex ℝ C ∧ C ⊆ K + K ∧
    (0.01 : ℝ≥0∞) ≤ gaussianMeasureInf C) := by
…
```
Mentions: Green54.gaussianMeasureInf