Erdős Problem 229

Reference: erdosproblems.com/229

theorem erdos_229 : (∀ (S : ℕ → Set ℂ), (∀ (n : ℕ), derivedSet (S n) = ∅) → ∃ (f : ℂ → ℂ), Transcendental (Polynomial ℂ) f ∧ Differentiable ℂ f ∧ ∀ n ≥ 1, ∃ (k : ℕ), ∀ z ∈ S n, iteratedDeriv k f z = 0) ↔ True

Let $(S_n)_{n \geq 1}$ 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 $n \geq 1$, there exists some $k_n \geq 0$ such that $$ f^{(k_n)}(z) = 0\quad\text{for all $z\in S_n$?} $$

Solved in the affirmative by Barth and Schneider [BaSc72].

[BaSc72] Barth, K. F. and Schneider, W. J., On a problem of Erdős concerning the zeros of the derivatives of an entire function. Proc. Amer. Math. Soc. (1972), 229--232.

theorem theorem_1 {S : ℕ → Set ℂ} (h : ∀ (k : ℕ), derivedSet (S k) = ∅) : ∃ (f : ℂ → ℂ) (n : ℕ → ℕ), Differentiable ℂ f ∧ Transcendental (Polynomial ℂ) f ∧ ∀ (k : ℕ), 0 < n k ∧ ∀ {z : ℂ}, z ∈ S k → iteratedDeriv (n k) f z = 0

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 $z \in S_k$.