Is there an entire non-zero function f:C→ C such that, for any infinite sequence n_1<n_2<…, the set z: f^(n_k)(z)=0 textrm for some k≥ 1 is everywhere dense?

Erdős Problem #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 den…

About 70 years old · traced to 1956

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?

References

Primary source

ULAM AI, UnsolvedMath dataset, Erdős Problems set, revision b9437975f3c873f635a13c48f8b022f5ba80898a; CC BY 4.0.

Additional references

UnsolvedMath, Erdős Problems set, ULAM AI, licensed CC BY 4.0.

analysis iterated-functions

InErdős Problems

Posted 1 month ago · edited 3 weeks ago

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

No solutions have been posted yet.