Vela Mathematics Program: problems

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79 problems · 1/2Number SourcesGraph view

declared falsifiable · $250erdos:114

declared proved (Lean) · formalizederdos:115

True ↔ ∀ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (p : Polynomial ℂ), p.Monic → p.natDegree = n → IsConnected {z | ‖Polynomial.eval z p‖ ≤ 1} → ∀ (z : ℂ), ‖Polynomial.eval z p‖ ≤ 1 → ‖Polynomial.eval z (Polynomial.derivative p)‖ ≤ (1 / 2 + ε) * ↑n ^ 2

polynomialsanalysisN/A5 sources

declared provederdos:116

Erdős problem 116

polynomialsanalysisN/A

declared solved · formalized · $100erdos:119

For unit-modulus complex numbers

z_i

, let

p_n(z)=\prod_{i\le n}(z-z_i)

and

M_n=\max_{|z|=1}|p_n(z)|

. Erdős's prize question: is there

c>0

with

\sum_{k\le n} M_k > n^{1+c}

analysispolynomialsN/A4 sources

declared provederdos:225

Erdős problem 225

analysisN/A

declared proved (Lean) · formalizederdos:226

True ↔ ∃ F, Differentiable ℂ F ∧ (∀ (x : ℝ), (F ↑x).im = 0) ∧ (∀ (g : ℝ →ᵃ[ℝ] ℝ), (fun x => (F ↑x).re) ≠ ⇑g) ∧ Erdos226.PreservesRationality fun x => (F ↑x).re

analysisN/A5 sources

declared disprovederdos:227

Erdős problem 227

analysisN/A

declared proved · formalizederdos:228

True ↔ ∃ c₁ c₂, ∀ᶠ (n : ℕ) in Filter.atTop, ∃ p, p.degree = ↑n ∧ (∀ i ≤ n, p.coeff i = 1 ∨ p.coeff i = -1) ∧ ∀ (z : ℂ), ‖z‖ = 1 → √↑n < c₁ * ‖Polynomial.eval z p‖ ∧ ‖Polynomial.eval z p‖ < c₂ * √↑n

analysispolynomialsN/A2 sources

declared proved (Lean) · formalizederdos:229

True ↔ ∀ (S : ℕ → Set ℂ), (∀ (n : ℕ), derivedSet (S n) = ∅) → ∃ f, Transcendental (Polynomial ℂ) f ∧ Differentiable ℂ f ∧ ∀ n ≥ 1, ∃ k, ∀ z ∈ S n, iteratedDeriv k f z = 0

analysisiterated functionsN/A5 sources

declared disprovederdos:230

Erdős problem 230

analysispolynomialsN/A

declared openerdos:256

Erdős problem 256

analysisN/A

declared provederdos:395

Erdős problem 395

analysisN/A

declared provederdos:485

Erdős problem 485

analysispolynomialspossible2 sources

declared proved · formalizederdos:494

∀ (k card p : ℕ), Nat.Prime p → k ∈ Set.Ioo 0 p → p ∣ card → Erdos494.Erdos494Unique k card

analysisadditive combinatoricsN/A2 sources

declared proved (Lean) · formalizederdos:498

True ↔ ∀ (n : ℕ) (z : Fin n → ℂ), (∀ (i : Fin n), 1 ≤ ‖z i‖) → ∀ (c : ℂ), {ε | (∀ (i : Fin n), ε i = -1 ∨ ε i = 1) ∧ ∑ i, ↑(ε i) * z i ∈ Metric.ball c 1}.ncard ≤ n.choose (n / 2)

combinatoricsanalysisN/A5 sources

declared open · formalizederdos:509

∀ {M : Type u} [inst : MetricSpace M] (S : Set M), S.Nonempty → ∀ (r : ℝ) (ι : Type v) (bdc : Erdos509.BoundedDiscCover S r ι), 0 < r

analysisN/A2 sources

declared open · formalizederdos:510

sorry ↔ ∃ c, ∃ (_ : 0 < c), ∀ᶠ (N : ℕ) in Filter.atTop, ∀ (A : Finset ℕ), 0 ∉ A → A.card = N → ∃ θ, ∑ n ∈ A, Real.cos (↑n * θ) < -c * √↑N

analysisN/A2 sources

declared disprovederdos:511

Erdős problem 511

analysisN/A

declared proved (Lean) · formalizederdos:512

True ↔ ∃ c > 0, ∀ (N : ℕ) (A : Finset ℤ), A.card = N → c * Real.log ↑N ≤ ∫ (θ : ℝ) in 0..1, ‖∑ n ∈ A, additiveChar (↑n * θ)‖

analysisN/A4 sources

declared open · formalizederdos:513

sorry = ⨆ f, Filter.liminf (fun r => Erdos513.ratio r ↑f) Filter.atTop

analysisN/A3 sources

declared openerdos:514

For a transcendental entire function, how fast can

|f(z)|

be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus

M(r, f)

analysisN/A4 sources

declared provederdos:515

Erdős problem 515

analysisN/A

declared proved · formalizederdos:516

∀ {f : ℂ → ℂ} {n : ℕ → ℕ}, HasFabryGaps n → ∀ {a : ℕ → ℂ}, (∀ (n : ℕ), a n ≠ 0) → (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) → Erdos516.OfFiniteOrder f → Filter.limsup (fun r => Erdos516.ratio r f) Filter.atTop = 1

analysisN/A2 sources

declared open · formalizederdos:517

sorry ↔ ∀ {f : ℂ → ℂ} {n : ℕ → ℕ}, HasFabryGaps n → ∀ {a : ℕ → ℂ}, (∀ (k : ℕ), a k ≠ 0) → (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) → ∀ (z : ℂ), {x | f x = z}.Infinite

analysisN/A3 sources

declared proved (Lean) · formalizederdos:519

True ↔ ∃ c, 0 < c ∧ ∀ (n : ℕ) (hn : 0 < n) (z : Fin n → ℂ), z ⟨0, hn⟩ = 1 → ∃ k, c < ‖Erdos519.powerSum z (↑k + 1)‖

analysisN/A5 sources

declared open · formalizederdos:521

False ↔ Erdos521.Claim

analysispolynomialsprobabilityN/A4 sources

declared open · formalizederdos:522

For

P_n(z) = \sum_{k=0}^n \varepsilon_k z^k

with independent uniform signs, does the number

R_n

of roots in

|z| \le 1

satisfy

R_n/(n/2) \to 1

almost surely? The manuscript proves the strong law with

R_n = n/2 + O_\omega(n^{149/150})

analysispolynomialsprobabilityN/A4 sources

declared provederdos:523

Erdős problem 523

analysisprobabilitypolynomialsN/A

declared openerdos:524

Erdős problem 524

analysisprobabilitypolynomialsN/A3 sources

declared provederdos:525

Erdős problem 525

analysisprobabilitypolynomialspossible

declared provederdos:527

Erdős problem 527

analysisprobabilityN/A

declared open · $250erdos:671

For triangular arrays of nodes

a_i^n\in[-1,1]

let

\mathcal{L}^nf

be the Lagrange interpolation polynomials of a continuous

f

, with fundamental polynomials

p_i^n

. Is there a choice of nodes such that for every continuous

f

there is some

x

where

\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty

and yet

\mathcal{L}^nf(x) \to f(x)

? Is there a choice with

\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty

for every

x

, yet for every continuous

f

some

x

has

\mathcal{L}^nf(x)\to f(x)

? Both questions are claimed resolved in the affirmative.

analysisN/A3 sources

declared open · formalizederdos:906

Is there an entire non-zero function

f:\mathbb{C}\to \mathbb{C}

such that, for any infinite sequence

n_1<n_2<\cdots

, the set

\{ z: f^{(n_k)}(z)=0 \textrm{ for some }k\geq 1\}

is everywhere dense? The literal question is trivial for polynomials, so the claims address the transcendental entire case, in the affirmative.

analysisiterated functionsN/A5 sources

declared proved (Lean) · formalizederdos:907

True ↔ ∀ (f : ℝ → ℝ), (∀ (h : ℝ), 0 < h → Continuous fun x => f (x + h) - f x) → ∃ g a, Continuous g ∧ (∀ (x y : ℝ), a (x + y) = a x + a y) ∧ f = g + a

analysisN/A5 sources

declared provederdos:908

Erdős problem 908

analysisN/A

declared provederdos:909

Erdős problem 909

analysistopologyN/A

declared disproved (Lean) · formalizederdos:967

False ↔ ∀ (a : ℕ → ℕ), StrictMono a → 1 < a 0 → (Summable fun k => 1 / ↑(a k)) → ∀ (t : ℝ), 1 + ∑' (k : ℕ), Erdos967.summand t (a k) ≠ 0

number theoryanalysisN/A5 sources

declared open · formalizederdos:973

True ↔ ∃ C > 1, ∀ n ≥ 2, ∃ z, z 1 = 1 ∧ (∀ i ∈ Finset.Icc 1 n, 1 ≤ ‖z i‖) ∧ ∀ k ∈ Finset.Icc 2 (n + 1), ‖∑ i ∈ Finset.Icc 1 n, z i ^ k‖ < C ^ (-↑n)

analysisN/A2 sources

declared proved (Lean) · formalizederdos:974

∀ {n : ℕ} [inst : NeZero n] (z : Fin n → ℂ), z 0 = 1 → {k | ∀ j < n - 1, ∑ i, z i ^ (k + j) = 0}.Infinite → Erdos974.IsTuranConfiguration z

analysisN/A5 sources

declared proved · formalizederdos:987

True ↔ ∀ (x : ℕ → ℝ), (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) → Filter.limsup (fun k => Erdos987.A x k) Filter.atTop = ⊤

analysisdiscrepancyN/A4 sources

declared disproved (Lean) · formalizederdos:990

False ↔ ∃ C, ∀ (f : Polynomial ℂ), f.coeff 0 ≠ 0 → ∀ (α β : ℝ), 0 ≤ α → α ≤ β → β ≤ 2 * Real.pi → |↑(Erdos990.rootArgCount f (Set.Icc α β)) - (β - α) / (2 * Real.pi) * ↑f.natDegree| ≤ C * √(↑f.support.card * Real.log (Erdos990.M f))

analysisN/A6 sources

declared disprovederdos:994

Erdős problem 994

analysisdiscrepancyN/A

declared openerdos:995

Erdős problem 995

analysisdiscrepancyN/A2 sources

declared open · formalizederdos:996

Let

n_1<n_2<\cdots

be a lacunary sequence of integers and

f\in L^2([0,1])

with

n

th Fourier partial sum

f_n

. Is there an absolute constant

C>0

such that if

\| f-f_n\|_2 \ll (\log\log\log n)^{-C}

then

\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})\to\int_0^1 f

for almost every

\alpha

? A preprint answers this negatively via a dyadic spike-block counterexample.

analysisN/A4 sources

declared proved (Lean) · formalizederdos:997

True ↔ ∀ (α : ℝ), ¬Erdos997.IsWellDistributed fun n => Int.fract (α * ↑(Nat.nth Nat.Prime n))

analysisdiscrepancyprimesN/A6 sources

declared provederdos:998

Erdős problem 998

analysisdiophantine approximationN/A

declared open · formalizederdos:1002

sorry ↔ ∃ g, Monotone g ∧ Filter.Tendsto g Filter.atBot (nhds 0) ∧ Filter.Tendsto g Filter.atTop (nhds 1) ∧ ∀ (c : ℝ), Filter.Tendsto (fun n => (MeasureTheory.volume {α | α ∈ Set.Ioo 0 1 ∧ (fun α n => 1 / Real.log ↑n * ∑ k ∈ Finset.Icc 1 n, (1 / 2 - Int.fract (α * ↑k))) α n ≤ c}).toReal) Filter.atTop (nhds (g c))

analysisdiophantine approximationN/A2 sources

declared open · formalizederdos:1038

Among all nonconstant monic polynomials

f

whose roots lie in

[-1, 1]

, determine

\inf_f |\{x \in \mathbb{R} : |f(x)| < 1\}|

analysisN/A4 sources

declared openerdos:1039

For

f(z) = \prod_{i=1}^n (z - z_i)

with all

|z_i| \le 1

, let

\rho(f)

be the radius of the largest disc contained in

\{z : |f(z)| < 1\}

. Is

\rho(f) \gg 1/n

? The worst case is now known to be

\Theta(1/n)

, with the explicit bound

\rho(f) \ge (\log 2)/n

analysisN/A3 sources

declared openerdos:1040

For a closed infinite set

F \subseteq \mathbb{C}

, let

\mu(F)

be the infimum of

|\{z : |f(z)| < 1\}|

over monic polynomials with zeros in

F

. Is

\mu(F)

determined only by the transfinite diameter of

F

analysisN/A3 sources

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