Erdős problem 1133 · problems.science

Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below $(1+\varepsilon)n$ to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.

Sources

Browse retained paths and inspect the exact material available for this Problem.

3 retained statements2415f78e850a

Open selected source

FormalConjectures/ErdosProblems/

1133.lean

Retained formal statement1 of 2

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 $\max_{x\in [-1,1]}\lvert P(x)\rvert >C.$

FormalConjectures/ErdosProblems/1133.leanErdos1133.erdos_113310 linesExact file

1True ↔2  ∀ C > 0,3    ∃ ε > 0,4      ∀ᶠ (n : ℕ) in Filter.atTop,5        ∀ (x : Fin n → ↑(Set.Icc (-1) 1)),6          ∃ y,7            ∀ (P : Polynomial ℝ),8              ↑P.natDegree < (1 + ε) * ↑n →9                ↑{i | Polynomial.eval (↑(x i)) P = ↑(y i)}.card ≥ (1 - ε) * ↑n →10                  ∃ z ∈ Set.Icc (-1) 1, |Polynomial.eval z P| > C

OpenStatement only, no proof

Sources

Search problems.science