Erdős problem 229

Let $(S_n)_{n \ge 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n \ge 1$ , there exists some $k_n \ge 0$ such that $f^{(k_n)}(z) = 0$ for all $z \in S_n$ .

Sources

The source's own material, retained exactly.

2 retained statements36c6d0fef71a

FormalConjectures/ErdosProblems/

229.lean

Retained formal statement2 of 2

Let $\{S_k\}$ be any sequence of sets in the complex plane, each of which has no finite limit point. Then there exists a sequence $\{n_k\}$ of positive integers and a transcendental entire function $f(z)$ such that $f^{(n_k)}(z) = 0$ if $z \in S_k$ .

FormalConjectures/ErdosProblems/229.leanErdos229.theorem_15 linesExact file

1∀ {S : ℕ → Set ℂ},2  (∀ (k : ℕ), derivedSet (S k) = ∅) →3    ∃ f n,4      Differentiable ℂ f ∧5        Transcendental (Polynomial ℂ) f ∧ ∀ (k : ℕ), 0 < n k ∧ ∀ {z : ℂ}, z ∈ S k → iteratedDeriv (n k) f z = 0

SolvedStatement only, no proof