FormalConjectures.ErdosProblems.«1133»

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import FormalConjecturesUtil

Erdős Problem 1133

References:

erdosproblems.com/1133

[Er67] Erdős, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73.


open Filter Set


namespace Erdos1133

Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds.

For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\ldots,y_n\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\epsilon)n$ many $1\leq i\leq n$, then $$\max_{x\in [-1,1]}\lvert P(x)\rvert >C.$$

@[category research open, AMS 26 41]
theorem declaration uses 'sorry'erdos_1133 :
    answer(sorry) ↔
    ∀ C > (0 : ℝ), ∃ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop,
      ∀ x : Fin n → Icc (-1 : ℝ) 1,
        ∃ y : Fin n → Icc (-1 : ℝ) 1,
          ∀ P : Polynomial ℝ,
            (P.natDegree : ℝ) < (1 + ε) * (n : ℝ) →
            ((Finset.univ.filter (fun i ↦ P.eval (x i : ℝ) = (y i : ℝ))).card : ℝ) ≥
              (1 - ε) * (n : ℝ) →
            ∃ z ∈ Icc (-1 : ℝ) 1, |P.eval z| > C := by⊢ True ↔
  ∀ C > 0,
    ∃ ε > 0,
      ∀ᶠ (n : ℕ) in atTop,
        ∀ (x : Fin n → ↑(Icc (-1) 1)),
          ∃ y,
            ∀ (P : Polynomial ℝ),
              ↑P.natDegree < (1 + ε) * ↑n →
                ↑{i | Polynomial.eval (↑(x i)) P = ↑(y i)}.card ≥ (1 - ε) * ↑n →
                  ∃ z ∈ Icc (-1) 1, |Polynomial.eval z P| > C
  sorryAll goals completed! 🐙

Erdős proved that, for any $C>0$, there exists $\epsilon>0$ such that if $n$ is sufficiently large and $m=\lfloor (1+\epsilon)n\rfloor$ then for any $x_1,\ldots,x_m\in [-1,1]$ there is a polynomial $P$ of degree $n$ such that $\lvert P(x_i)\rvert\leq 1$ for $1\leq i\leq m$ and $\max_{x\in [-1,1]}\lvert P(x)\rvert>C$. The conjectured statement would also imply this, but Erdős in [Er67] says he could not even prove it for $m=n$.

@[category research solved, AMS 26 41]
theorem declaration uses 'sorry'erdos_1133.variants.weaker :
    ∀ C > (0 : ℝ), ∃ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop,
      let m := ⌊(1 + ε) * (n : ℝ)⌋₊;
      ∀ x : Fin m → Icc (-1 : ℝ) 1,
        ∃ P : Polynomial ℝ,
          P.natDegree = n ∧
          (∀ i : Fin m, |P.eval (x i : ℝ)| ≤ 1) ∧
          ∃ z ∈ Icc (-1 : ℝ) 1, |P.eval z| > C := by⊢ ∀ C > 0,
  ∃ ε > 0,
    ∀ᶠ (n : ℕ) in atTop,
      let m := ⌊(1 + ε) * ↑n⌋₊;
      ∀ (x : Fin m → ↑(Icc (-1) 1)),
        ∃ P,
          P.natDegree = n ∧
            (∀ (i : Fin m), |Polynomial.eval (↑(x i)) P| ≤ 1) ∧ ∃ z ∈ Icc (-1) 1, |Polynomial.eval z P| > C
  sorryAll goals completed! 🐙


end Erdos1133