Research Problems in Function Theory — Problem 2.42

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An asymptotic path is a continuous curve $\gamma:[0,\infty)\to\mathbb C$ with $|\gamma(t)|\to\infty$. An entire function has asymptotic value $a$ along it if $f(\gamma(t))\to a$. Let $n(r,a,f)$ count all $a$-points in $|z|\le r$ with multiplicity, and let $n_1(r,a,f,\gamma)$ count those lying on the trace of $\gamma$, counted once with their zero multiplicity even if the curve visits them repeatedly.

Statement recorded in the supplied source.

For $l\ge2$, can one construct an entire function $f$, distinct finite asymptotic values $a_1,\ldots,a_l$, and corresponding asymptotic paths $\gamma_1,\ldots,\gamma_l$, with $n(r,a_k,f)>0$ for all sufficiently large $r$ and every $k$, such that \[\lim_{r\to\infty}\frac{n_1(r,a_k,f,\gamma_k)}{n(r,a_k,f)}=b_k>0 \quad(k=1,\ldots,l)?\] Can all the constants $b_k$ be taken to equal $1$? [S2] Update 2.42 reports Barsegyan's constraint $\sum_k b_k\le1$ for entire functions, so the all-one alternative is excluded when $l\ge2$. [S2]

Short English statement

Can a positive fraction of the points taking each of several asymptotic values lie on its asymptotic path, and can each such fraction be one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302042 (AMR-022-2042). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.42 and Update 2.42. https://arxiv.org/abs/1809.07200.