959.lean`

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

10 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 959

Reference: erdosproblems.com/959

Imports (1)

FormalConjecturesUtil

def Erdos959.indexPairs The unordered index pairs, represented uniquely by the orientation i < j line 35 source
Source documentation
The unordered index pairs, represented uniquely by the orientation i < j.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Prod, Fin
Proof uses: Finset.filter, LT.lt, Prod.fst, Prod.snd, SProd.sprod, Finset.univ
4 instances, coercions and auxiliary constants left out.
```
def indexPairs (n : ℕ) : Finset (Fin n × Fin n) :=
…
```
def Erdos959.distanceValues The finite set of distinct distances determined by a configuration line 39 source
Source documentation
The finite set of distinct distances determined by a configuration.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, EuclideanSpace, Real, OfNat.ofNat, Finset
Proof uses: Finset.image, Prod, Dist.dist, ENNReal, Prod.fst, Prod.snd, Erdos959.indexPairs
12 instances, coercions and auxiliary constants left out.
```
def distanceValues {n : ℕ} (P : Fin n → ℝ²) : Finset ℝ :=
…
```
def Erdos959.frequency $f(d)$: the number of unordered pairs in $P$ determining distance $d$ line 43 source
Source documentation
$f(d)$: the number of unordered pairs in $P$ determining distance $d$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, EuclideanSpace, Real, OfNat.ofNat
Proof uses: Finset.card, Prod, Finset.filter, Eq, Dist.dist, ENNReal, Prod.fst, Prod.snd, Erdos959.indexPairs
12 instances, coercions and auxiliary constants left out.
```
def frequency {n : ℕ} (P : Fin n → ℝ²) (d : ℝ) : ℕ :=
…
```
def Erdos959.runnerUpFrequency For a fixed distance $d$, the largest multiplicity among all other represented distances (Finset.sup returns $0$ when there is no other value) line 48 source
Source documentation
For a fixed distance $d$, the largest multiplicity among all other represented distances (Finset.sup returns $0$ when there is no other value).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, EuclideanSpace, Real, OfNat.ofNat
Proof uses: Finset.sup, Finset.erase, Erdos959.distanceValues, Erdos959.frequency
6 instances, coercions and auxiliary constants left out.
```
def runnerUpFrequency {n : ℕ} (P : Fin n → ℝ²) (d : ℝ) : ℕ :=
…
```
def Erdos959.multiplicityGap $f(d_1) - f(d_2)$: the gap between the largest and second-largest distance multiplicities line 55 source
Source documentation
$f(d_1) - f(d_2)$: the gap between the largest and second-largest distance multiplicities. The supremum over $d$ handles both a unique winner and a tie: non-winners contribute $0$ by truncated subtraction, and tied winners also contribute $0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, EuclideanSpace, Real, OfNat.ofNat
Proof uses: Finset.sup, Erdos959.distanceValues, HSub.hSub, Erdos959.frequency, Erdos959.runnerUpFrequency
7 instances, coercions and auxiliary constants left out.
```
def multiplicityGap {n : ℕ} (P : Fin n → ℝ²) : ℕ :=
…
```
def Erdos959.Admissible A configuration represents a set of size $n$ (injective) and has at least two distinct distances, so that $d_2$ exists line 60 source
Source documentation
A configuration represents a set of size $n$ (injective) and has at least two distinct distances, so that $d_2$ exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, EuclideanSpace, Real, OfNat.ofNat
Proof uses: And, Function.Injective, LE.le, Finset.card, Erdos959.distanceValues
3 instances, coercions and auxiliary constants left out.
```
def Admissible {n : ℕ} (P : Fin n → ℝ²) : Prop :=
…
```
def Erdos959.AttainableGap A natural number occurs as the top-two multiplicity gap of some admissible $n$-point configuration line 65 source
Source documentation
A natural number occurs as the top-two multiplicity gap of some admissible $n$-point configuration.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Fin, EuclideanSpace, Real, OfNat.ofNat, And, Erdos959.Admissible, Eq, Erdos959.multiplicityGap
1 instances, coercions and auxiliary constants left out.
```
def AttainableGap (n g : ℕ) : Prop :=
…
```
def Erdos959.extremalGap The extremal quantity of the problem, $\max_{|A| = n} (f(d_1) - f(d_2))$ line 70 source
Source documentation
The extremal quantity of the problem, $\max_{|A| = n} (f(d_1) - f(d_2))$. There are Nat.choose n 2 unordered pairs, so this is a valid finite search interval.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.findGreatest, Erdos959.AttainableGap, Classical.propDecidable, Nat.choose, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def extremalGap (n : ℕ) : ℕ := by
…
```
theorem Erdos959.erdos_959 Let $A\subseteq \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$ sorry line 83 source
Source documentation
Let $A\subseteq \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$ is determined, ordered so that $f(d_1)\geq f(d_2)\geq \cdots \geq f(d_k)$. Estimate $$\max (f(d_1)-f(d_2)),$$ where the maximum is taken over all $A$ of size $n$ (this is extremalGap n).

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos959.extremalGap, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_959 :
    (fun n ↦ (extremalGap n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: Erdos959.extremalGap
theorem Erdos959.erdos_959.lower_bound Documented declaration: lower_bound sorry line 94 source
Source documentation
A superlinear lower bound: there is a $c>0$ with $$\text{extremalGap}(n)\geq n^{1 + c/\log\log n}$$ for all large $n$, so $\max (f(d_1)-f(d_2))$ grows faster than any linear function of $n$. Determining the exact order remains open, which is erdos_959.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, GE.ge, LE.le, HPow.hPow, HAdd.hAdd, HDiv.hDiv, Real.log, Erdos959.extremalGap
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
33 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_959.lower_bound :
    ∃ c : ℝ, 0 < c ∧ ∃ N : ℕ, ∀ n ≥ N,
      (n : ℝ) ^ (1 + c / Real.log (Real.log n)) ≤ extremalGap n := by
…
```
Mentions: c, Erdos959.extremalGap