466.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 466

References:

erdosproblems.com/466
[Er72] Erdős, Paul, Extremal problems in number theory. Proceedings of the 1972 Number Theory Conference (Univ. Colorado, Boulder, Colo.) (1972), 80-86.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59-79.
[Sa76] Sárközy, A., On distances near integers. I, II. Studia Sci. Math. Hungar. (1976), 37-50, 105-111.

Imports (1)

FormalConjecturesUtil

def Erdos466.N N X δ is the maximum number of points in a closed disc of radius X in the plane such that the distance between any two of them is at least δ away from the nearest integer line 45 source
Source documentation
N X δ is the maximum number of points in a closed disc of radius X in the plane such that the distance between any two of them is at least δ away from the nearest integer.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, EuclideanSpace, Fin, OfNat.ofNat, Finset, And, Eq, Finset.card, LE.le, Set, Metric.closedBall, ENNReal, Set.Pairwise, distToNearestInt, Dist.dist
17 instances, coercions and auxiliary constants left out.
```
noncomputable def N (X δ : ℝ) : ℕ :=
…
```
theorem Erdos466.erdos_466 sorry line 65 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Tendsto, Nat, Erdos466.N, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos466.erdos_466.variants.graham Graham proved that $N(X,1/10)> \frac{\log X}{10}$ sorry line 70 source
Source documentation
Graham proved that $N(X,1/10)> \frac{\log X}{10}$.

depends on sorry what this means
Statement uses: Real, LE.le, OfNat.ofNat, LT.lt, HDiv.hDiv, Real.log, Nat, Erdos466.N
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_466.variants.graham : ∀ X : ℝ, 1 ≤ X → Real.log X / 10 < N X (1 / 10) := by
…
```
Mentions: Erdos466.N
theorem Erdos466.erdos_466.variants.sarkozy Sárközy [Sa76] proved that for all sufficiently small $\delta>0$, $N(X,\delta)>X^{1/2-\delta^{1/7}}$ for all sufficiently large $X$ sorry line 78 source
Source documentation
Sárközy [Sa76] proved that for all sufficiently small $\delta>0$, $N(X,\delta)>X^{1/2-\delta^{1/7}}$ for all sufficiently large $X$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, HPow.hPow, HSub.hSub, HDiv.hDiv, Nat, Erdos466.N, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_466.variants.sarkozy : ∃ δ₀ : ℝ, 0 < δ₀ ∧ ∀ δ : ℝ, 0 < δ → δ < δ₀ →
    ∀ᶠ X : ℝ in atTop, X ^ (1 / 2 - δ ^ (1 / 7 : ℝ)) < N X δ := by
…
```
Mentions: Erdos466.N