769.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 769

Reference: erdosproblems.com/769

Imports (1)

FormalConjecturesUtil

structure Erdos769.Cube An axis-parallel homothetic copy of the standard n-cube line 35 source
Source documentation
An axis-parallel homothetic copy of the standard n-cube.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
```
structure Cube (n : ℕ) where
…
```
def Erdos769.Cube.Mem Membership in the half-open cube ∏ j, [lower j, lower j + side) line 40 source
Source documentation
Membership in the half-open cube ∏ j, [lower j, lower j + side).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Erdos769.Cube, Fin, Real
Proof uses: And, LE.le, Erdos769.Cube.lower, LT.lt, HAdd.hAdd, Erdos769.Cube.side
4 instances, coercions and auxiliary constants left out.
```
def Cube.Mem {n : ℕ} (Q : Cube n) (x : Fin n → ℝ) : Prop :=
…
```
Mentions: Erdos769.Cube
def Erdos769.InUnit Membership in the half-open unit cube [0,1)^n line 44 source
Source documentation
Membership in the half-open unit cube [0,1)^n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Real
Proof uses: And, LE.le, OfNat.ofNat, LT.lt
6 instances, coercions and auxiliary constants left out.
```
def InUnit {n : ℕ} (x : Fin n → ℝ) : Prop :=
…
```
def Erdos769.Cube.InsideUnit A cube is nondegenerate and contained in the closed unit cube line 48 source
Source documentation
A cube is nondegenerate and contained in the closed unit cube.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Erdos769.Cube
Proof uses: And, LT.lt, Real, OfNat.ofNat, Erdos769.Cube.side, Fin, LE.le, Erdos769.Cube.lower, HAdd.hAdd
8 instances, coercions and auxiliary constants left out.
```
def Cube.InsideUnit {n : ℕ} (Q : Cube n) : Prop :=
…
```
Mentions: Erdos769.Cube
def Erdos769.IsTiling Documented declaration: IsTiling line 54 source
Source documentation
tiles is an exact decomposition of the unit n-cube into k homothetic cubes: every tile is a positive homothet inside the unit cube, and every point of [0,1)^n belongs to exactly one half-open tile.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Erdos769.Cube
Proof uses: And, Erdos769.Cube.InsideUnit, Real, Erdos769.InUnit, ExistsUnique, Erdos769.Cube.Mem
```
def IsTiling {n k : ℕ} (tiles : Fin k → Cube n) : Prop :=
…
```
Mentions: Erdos769.Cube
def Erdos769.Admissible Exactly k homothetic n-cubes can tile the unit n-cube line 58 source
Source documentation
Exactly k homothetic n-cubes can tile the unit n-cube.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Fin, Erdos769.Cube, Erdos769.IsTiling
```
def Admissible (n k : ℕ) : Prop :=
…
```
def Erdos769.IsCutoff c is the minimal cutoff c(n): every k ≥ c is admissible, and (when c > 0) c - 1 is not line 63 source
Source documentation
c is the minimal cutoff c(n): every k ≥ c is admissible, and (when c > 0) c - 1 is not.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, LE.le, Erdos769.Admissible, Or, Eq, OfNat.ofNat, Not, HSub.hSub
4 instances, coercions and auxiliary constants left out.
```
def IsCutoff (n c : ℕ) : Prop :=
…
```
Mentions: c
def Erdos769.Erdos769LowerBound The highlighted conjecture c(n) ≫ n^n: there is an absolute positive constant A/B with c(n) ≥ (A/B) n^n for all sufficiently large n line 68 source
Source documentation
The highlighted conjecture c(n) ≫ n^n: there is an absolute positive constant A/B with c(n) ≥ (A/B) n^n for all sufficiently large n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Exists, Nat, And, LT.lt, OfNat.ofNat, LE.le, Erdos769.IsCutoff, HMul.hMul, HPow.hPow
9 instances, coercions and auxiliary constants left out.
```
def Erdos769LowerBound : Prop :=
…
```
theorem Erdos769.erdos_769 Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes sorry line 84 source
Source documentation
Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ — in particular, is it true that $c(n)\gg n^n$?
The c(n) \gg n^n conjecture is false: for odd n, one can tile the unit n-cube into k homothetic cubes for every k ≥ U(n) with U(n)/n^n → 0, so c(n)/n^n → 0 along odd dimensions and no absolute constant lower-bounds it.
Determining good bounds for c(n) in general remains open.

depends on sorry what this means
Statement uses: Iff, False, Erdos769.Erdos769LowerBound
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_769 : answer(False) ↔ Erdos769LowerBound := by
…
```
Mentions: Erdos769.Erdos769LowerBound
theorem Erdos769.erdos_769.variants.growth_rate What refuting $c(n)\gg n^n$ leaves open: give good bounds for $c(n)$ sorry line 96 source
Source documentation
What refuting $c(n)\gg n^n$ leaves open: give good bounds for $c(n)$. Asked here on the scale the problem itself sets, $n^n$: does $c$ have a well-defined order $$\lim_{n\to\infty}\frac{\log c(n)}{n\log n}?$$ The refuted conjecture would have forced this limit to be at least $1$. The cutoff condition is restricted to positive dimensions: in dimension zero, exact coverage permits only one tile, so no cutoff exists.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, LT.lt, OfNat.ofNat, Erdos769.IsCutoff, Real, Filter.Tendsto, HDiv.hDiv, Real.log, HMul.hMul, Filter.atTop, nhds
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_769.variants.growth_rate :
    answer(sorry) ↔
      ∃ c : ℕ → ℕ, (∀ n, 0 < n → IsCutoff n (c n)) ∧
        ∃ γ : ℝ, Tendsto (fun n : ℕ => Real.log (c n) / (n * Real.log n)) atTop (𝓝 γ) := by
…
```
Mentions: c, Erdos769.IsCutoff