525.lean`

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

9 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 525

References:

erdosproblems.com/525
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Li66] Littlewood, J. E., On polynomials $\sum^n \pm z^m$, $\sum^n e^{\alpha_m i} z^m$, $z = e^{\theta i}$. J. London Math. Soc. (1966), 367-376.
[Ka87] Kashin, B. S., The properties of random trigonometric polynomials with $\pm 1$ coefficients. Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1987), 40-46, 105.
[Ko94] Konyagin, S. V., On the minimum modulus of random trigonometric polynomials with coefficients $\pm1$. Mat. Zametki (1994), 80-101, 158.
[KoSc99] Konyagin, S. V. and Schlag, W., Lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle. Trans. Amer. Math. Soc. (1999), 4963-4980.
[CoNg21] Cook, Nicholas A. and Nguyen, Hoi H., Universality of the minimum modulus for random trigonometric polynomials. Discrete Anal. (2021), Paper No. 20, 46.

Imports (1)

FormalConjecturesUtil

def Erdos525.eval The degree-n polynomial $f(z)=\sum_{j\leq n}\epsilon_j z^j$ with coefficients ε : Fin (n + 1) → ℤˣ, i.e. $\epsilon_j\in\{-1,1\}$, evaluated at z line 47 source
Source documentation
The degree-n polynomial $f(z)=\sum_{j\leq n}\epsilon_j z^j$ with coefficients ε : Fin (n + 1) → ℤˣ, i.e. $\epsilon_j\in\{-1,1\}$, evaluated at z.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Units, Int, Complex
Proof uses: Finset.sum, Finset.univ, HMul.hMul, HPow.hPow
21 instances, coercions and auxiliary constants left out.
```
def eval {n : ℕ} (ε : Fin (n + 1) → ℤˣ) (z : ℂ) : ℂ :=
…
```
def Erdos525.exceptional The degree-n polynomials with $\pm 1$ coefficients such that $|f(z)|\geq 1$ for all $|z|=1$ line 52 source
Source documentation
The degree-n polynomials with $\pm 1$ coefficients such that $|f(z)|\geq 1$ for all $|z|=1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Fin, HAdd.hAdd, OfNat.ofNat, Units, Int
Proof uses: Set.ofPred, Complex, Eq, Real, Norm.norm, LE.le, Erdos525.eval
12 instances, coercions and auxiliary constants left out.
```
def exceptional (n : ℕ) : Set (Fin (n + 1) → ℤˣ) :=
…
```
def Erdos525.m $m(f)=\min_{|z|=1}|f(z)|$ line 56 source
Source documentation
$m(f)=\min_{|z|=1}|f(z)|$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Units, Int, Real
Proof uses: InfSet.sInf, Set.image, Complex, Norm.norm, Erdos525.eval, Metric.sphere
19 instances, coercions and auxiliary constants left out.
```
noncomputable def m {n : ℕ} (ε : Fin (n + 1) → ℤˣ) : ℝ :=
…
```
def Erdos525.prob The probability that a uniformly random degree-n polynomial with $\pm 1$ coefficients satisfies P line 62 source
Source documentation
The probability that a uniformly random degree-n polynomial with $\pm 1$ coefficients satisfies P.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Units, Int, Real
Proof uses: HDiv.hDiv, Finset.card, Finset.filter, Classical.propDecidable, Finset.univ, HPow.hPow
23 instances, coercions and auxiliary constants left out.
```
noncomputable def prob (n : ℕ) (P : (Fin (n + 1) → ℤˣ) → Prop) : ℝ :=
…
```
theorem Erdos525.erdos_525.parts.i sorry line 74 source
depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Set.ncard, Fin, HAdd.hAdd, OfNat.ofNat, Units, Int, Erdos525.exceptional, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos525.erdos_525.parts.ii sorry line 89 source
depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Tendsto, Nat, Erdos525.prob, Fin, HAdd.hAdd, Units, Int, HDiv.hDiv, Real.sqrt, Erdos525.m, Filter.atTop, nhds, Real.exp, HMul.hMul, Neg.neg, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos525.erdos_525.variants.kashin Littlewood [Li66] conjectured that $m(f)=o(1)$ holds almost surely; this was proved by Kashin [Ka87] sorry line 97 source
Source documentation
Littlewood [Li66] conjectured that $m(f)=o(1)$ holds almost surely; this was proved by Kashin [Ka87].

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Tendsto, Nat, Erdos525.prob, Fin, HAdd.hAdd, Units, Int, Erdos525.m, 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_525.variants.kashin (δ : ℝ) (hδ : 0 < δ) :
    Tendsto (fun n : ℕ ↦ prob n fun f ↦ m f < δ) atTop (nhds 1) := by
…
```
Mentions: Erdos525.prob, Erdos525.m
theorem Erdos525.erdos_525.variants.konyagin Konyagin [Ko94] proved that $m(f)\leq n^{-1/2+o(1)}$ almost surely sorry line 103 source
Source documentation
Konyagin [Ko94] proved that $m(f)\leq n^{-1/2+o(1)}$ almost surely.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Tendsto, Nat, Erdos525.prob, Fin, HAdd.hAdd, Units, Int, LE.le, Erdos525.m, HPow.hPow, HDiv.hDiv, Neg.neg, Filter.atTop, nhds
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_525.variants.konyagin (δ : ℝ) (hδ : 0 < δ) :
    Tendsto (fun n : ℕ ↦ prob n fun f ↦ m f ≤ (n : ℝ) ^ (-1 / 2 + δ : ℝ)) atTop
      (nhds 1) := by
…
```
Mentions: Erdos525.prob, Erdos525.m
theorem Erdos525.erdos_525.variants.konyagin_schlag Konyagin and Schlag [KoSc99] proved that for any $\epsilon>0$ $$\limsup_{n\to \infty} \mathbb{P}(m(f) \leq \epsilon n^{-1/2})\ll \epsilon.$$ sorry line 111 source
Source documentation
Konyagin and Schlag [KoSc99] proved that for any $\epsilon>0$ $$\limsup_{n\to \infty} \mathbb{P}(m(f) \leq \epsilon n^{-1/2})\ll \epsilon.$$

depends on sorry what this means
Statement uses: Exists, Real, LT.lt, OfNat.ofNat, LE.le, Filter.limsup, Nat, Erdos525.prob, Fin, HAdd.hAdd, Units, Int, Erdos525.m, HDiv.hDiv, Real.sqrt, Filter.atTop, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_525.variants.konyagin_schlag :
    ∃ C : ℝ, ∀ ε : ℝ, 0 < ε →
      limsup (fun n : ℕ ↦ prob n fun f ↦ m f ≤ ε / √n) atTop ≤ C * ε := by
…
```
Mentions: Erdos525.prob, Erdos525.m