1048.lean`

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

9 declarations, 8 contain sorry, 0 declared as axioms

Erdős Problem 1048

References:

erdosproblems.com/1048
[EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.

Imports (2)

def Erdos1048.openLevelSet The set $\{ z \in \mathbb{C} : \lvert f(z)\rvert < 1\}$ line 39 source
Source documentation
The set $\{ z \in \mathbb{C} : \lvert f(z)\rvert < 1\}$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex, Set
Proof uses: Set.ofPred, LT.lt, Real, Norm.norm, Polynomial.eval, OfNat.ofNat
6 instances, coercions and auxiliary constants left out.
```
def openLevelSet (f : Polynomial ℂ) : Set ℂ :=
…
```
theorem Erdos1048.erdos_1048 Documented declaration: erdos_1048 sorry line 55 source
Source documentation
If $f\in \mathbb{C}[x]$ is a monic polynomial with all roots satisfying $\lvert z\rvert \leq r$ for some $0 < r < 2$, then must $$\{ z: \lvert f(z)\rvert <1\}$$ have a connected component with diameter $>2-r$?
A problem of Erdős, Herzog, and Piranian [EHP58].
Pommerenke [Po61] proved the answer is no for $r>1$, showing that if $f(z)=z^n-r^n$ then $\{ z: \lvert f(z)\rvert \leq 1\}$ has $n$ connected components, all with diameter $\to 0$ as $n\to \infty$.

depends on sorry what this means
Statement uses: Iff, False, Real, Polynomial, Complex, LT.lt, OfNat.ofNat, Nat, Polynomial.Monic, GE.ge, WithBot, Polynomial.degree, Membership.mem, Multiset, Polynomial.roots, LE.le, Norm.norm, Exists, And, Set and 6 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
80 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1048 : answer(False) ↔
    ∀ (r : ℝ) (f : ℂ[X]), 0 < r → r < 2 → f.Monic → f.degree ≥ 1 → (∀ z ∈ f.roots, ‖z‖ ≤ r) →
      ∃ z ∈ openLevelSet f, ENNReal.ofReal (2 - r) <
        Metric.ediam (connectedComponentIn (openLevelSet f) z) := by
…
```
Mentions: Erdos1048.openLevelSet
theorem Erdos1048.erdos_1048.variants.pommerenke_ncard_components Documented declaration: pommerenke_ncard_components sorry line 67 source
Source documentation
Pommerenke [Po61] proved the answer is no for $r>1$, showing that if $f(z)=z^n-r^n$ then $\{ z: \lvert f(z)\rvert \leq 1\}$ has $n$ connected components, all with diameter $\to 0$ as $n\to \infty$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Nat, LE.le, Polynomial, Complex, Eq, HSub.hSub, HPow.hPow, Polynomial.X, RingHom, Polynomial.C, Set.ncard, Set, Set.ofPred, Exists, And, Membership.mem, Erdos1043.levelSet and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1048.variants.pommerenke_ncard_components (r : ℝ) (hr : 1 < r) (n : ℕ) (hn : 1 ≤ n)
    (f : ℂ[X]) (hf : f = (X : ℂ[X]) ^ n - C ((r : ℂ) ^ n)) :
    {C : Set ℂ | ∃ z ∈ levelSet f, C = connectedComponentIn (levelSet f) z}.ncard = n := by
…
```
theorem Erdos1048.erdos_1048.variants.pommerenke_diam_tendsto_zero Documented declaration: pommerenke_diam_tendsto_zero sorry line 78 source
Source documentation
Pommerenke [Po61] proved the answer is no for $r>1$, showing that if $f(z)=z^n-r^n$ then $\{ z: \lvert f(z)\rvert \leq 1\}$ has $n$ connected components, all with diameter $\to 0$ as $n\to \infty$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Polynomial, Complex, Eq, HSub.hSub, HPow.hPow, Polynomial.X, RingHom, Polynomial.C, Membership.mem, Set, Erdos1043.levelSet, ENNReal, Metric.ediam, connectedComponentIn, ENNReal.ofReal and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
69 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1048.variants.pommerenke_diam_tendsto_zero (r : ℝ) (hr : 1 < r) (ε : ℝ)
    (hε : 0 < ε) :
    ∀ᶠ n : ℕ in Filter.atTop, ∀ f : ℂ[X], f = (X : ℂ[X]) ^ n - C ((r : ℂ) ^ n) →
      ∀ z ∈ levelSet f,
        Metric.ediam (connectedComponentIn (levelSet f) z) < ENNReal.ofReal ε := by
…
```
theorem Erdos1048.erdos_1048.variants.r_le_one On the other hand, if $0<r\leq 1$, then the answer is yes, as also shown by Pommerenke [Po61] sorry line 89 source
Source documentation
On the other hand, if $0<r\leq 1$, then the answer is yes, as also shown by Pommerenke [Po61].

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, LE.le, Polynomial, Complex, Polynomial.Monic, GE.ge, WithBot, Nat, Polynomial.degree, Membership.mem, Multiset, Polynomial.roots, Norm.norm, Exists, And, Set, Erdos1048.openLevelSet, ENNReal and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
82 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1048.variants.r_le_one (r : ℝ) (hr₀ : 0 < r) (hr₁ : r ≤ 1) (f : ℂ[X])
    (hmonic : f.Monic) (hdeg : f.degree ≥ 1) (hroots : ∀ z ∈ f.roots, ‖z‖ ≤ r) :
    ∃ z ∈ openLevelSet f, ENNReal.ofReal (2 - r) <
      Metric.ediam (connectedComponentIn (openLevelSet f) z) := by
…
```
Mentions: Erdos1048.openLevelSet
theorem Erdos1048.erdos_1048.variants.diam_ge_two If $0\leq r\leq 1/2$ then the component which contains $0$ must have diameter $\geq 2$, which $f(z)=z^n$ shows is best possible sorry line 100 source
Source documentation
If $0\leq r\leq 1/2$ then the component which contains $0$ must have diameter $\geq 2$, which $f(z)=z^n$ shows is best possible.

depends on sorry what this means
Statement uses: Real, LE.le, OfNat.ofNat, HDiv.hDiv, Nat, Polynomial, Complex, Polynomial.Monic, GE.ge, WithBot, Polynomial.degree, Membership.mem, Multiset, Polynomial.roots, Norm.norm, ENNReal, Metric.ediam, connectedComponentIn, Erdos1043.levelSet
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
84 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1048.variants.diam_ge_two (r : ℝ) (hr₀ : 0 ≤ r) (hr₁ : r ≤ 1 / 2) (f : ℂ[X])
    (hmonic : f.Monic) (hdeg : f.degree ≥ 1) (hroots : ∀ z ∈ f.roots, ‖z‖ ≤ r) :
    2 ≤ Metric.ediam (connectedComponentIn (levelSet f) 0) := by
…
```
theorem Erdos1048.erdos_1048.variants.diam_ge_two_is_best If $0\leq r\leq 1/2$ then the component which contains $0$ must have diameter $\geq 2$, which $f(z)=z^n$ shows is best possible sorry line 110 source
Source documentation
If $0\leq r\leq 1/2$ then the component which contains $0$ must have diameter $\geq 2$, which $f(z)=z^n$ shows is best possible.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, ENNReal, Metric.ediam, Complex, connectedComponentIn, Erdos1043.levelSet, HPow.hPow, Polynomial, Polynomial.X
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1048.variants.diam_ge_two_is_best (n : ℕ) (hn : 1 ≤ n) :
    Metric.ediam (connectedComponentIn (levelSet ((X : ℂ[X]) ^ n)) 0) = 2 := by
…
```
theorem Erdos1048.erdos_1048.variants.diam_gt_inv_r If $1/21/r$ sorry line 119 source
Source documentation
If $1/2<r\leq \frac{\sqrt{5}-1}{2}$ then the component which contains $0$ must have diameter $>1/r$.

L4DEL explanation
If $1/21/r$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Real, LT.lt, HDiv.hDiv, OfNat.ofNat, Nat, LE.le, HSub.hSub, Real.sqrt, Polynomial, Complex, Polynomial.Monic, GE.ge, WithBot, Polynomial.degree, Membership.mem, Multiset, Polynomial.roots, Norm.norm, ENNReal, ENNReal.ofReal and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
86 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1048.variants.diam_gt_inv_r (r : ℝ) (hr₀ : 1 / 2 < r)
    (hr₁ : r ≤ (Real.sqrt 5 - 1) / 2) (f : ℂ[X]) (hmonic : f.Monic) (hdeg : f.degree ≥ 1)
    (hroots : ∀ z ∈ f.roots, ‖z‖ ≤ r) :
    ENNReal.ofReal (1 / r) < Metric.ediam (connectedComponentIn (levelSet f) 0) := by
…
```
theorem Erdos1048.erdos_1048.variants.diam_gt_two_sub_sq If $\frac{\sqrt{5}-1}{2}\leq r\leq 1$ then the component which contains $0$ must have diameter $>2-r^2$ sorry line 130 source
Source documentation
If $\frac{\sqrt{5}-1}{2}\leq r\leq 1$ then the component which contains $0$ must have diameter $>2-r^2$.

depends on sorry what this means
Statement uses: Real, LE.le, HDiv.hDiv, HSub.hSub, Real.sqrt, OfNat.ofNat, Nat, Polynomial, Complex, Polynomial.Monic, GE.ge, WithBot, Polynomial.degree, Membership.mem, Multiset, Polynomial.roots, Norm.norm, LT.lt, ENNReal, ENNReal.ofReal and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
92 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1048.variants.diam_gt_two_sub_sq (r : ℝ) (hr₀ : (Real.sqrt 5 - 1) / 2 ≤ r)
    (hr₁ : r ≤ 1) (f : ℂ[X]) (hmonic : f.Monic) (hdeg : f.degree ≥ 1)
    (hroots : ∀ z ∈ f.roots, ‖z‖ ≤ r) :
    ENNReal.ofReal (2 - r ^ 2) < Metric.ediam (connectedComponentIn (levelSet f) 0) := by
…
```