---

Claim. Both questions are answered. Let $f$ be a nonconstant entire function and $E(c)=\{z:|f(z)|>c\}$. First, the minimal growth is the one Hayman conjectured: if $E(c)$ has finite planar measure for some $c>0$, then

$$\int^\infty\frac{r\,dr}{\log\log M(r,f)}<\infty$$

(Part 1, formula (6)), and the bound is best possible: for every continuous positive nondecreasing $\Phi$ with $\int^\infty r\,dr/\Phi(r)<\infty$ there is an entire $f$ with $\log\log M(r,f)=O(\Phi(r))$ and $|E(c)|<\infty$ for every $c>0$ (Part 2, with a lemma credited to V. S. Boichuk). Second, the answer to Erdős's question is no: finiteness of $|E(c)|$ for one $c$ does not force finiteness of $|E(c')|$ for some $c'<c$. The paper says at the outset (p. 512) that it inserts the word "some" into Erdős's question because the answer is negative already in that form, and hence in the form asking for every $c'<c$. Part 3 (p. 517) gives the shape of the examples: with $T=\{c>0:|E(c)|<\infty\}$, the cases $T=\emptyset$ and $T=(0,\infty)$ are clear (the function of Part 2 has the latter), and for every $m>0$ there are entire functions with $T=[m,\infty)$ and with $T=(m,\infty)$, built by Keldysh approximation of $m(1-\tfrac14z^{-1/4})^2$, respectively $m(1+\tfrac14z^{-1/4})^2$, outside a domain of finite area; the site's commentary reports the same description. The proof of Part 1 applies the inequality proved independently by Pfluger and by Arima with Carleman's method, $\log^+\log^+M(er,f)\ge\pi\int_{A(r)}dt/l(t)-K$, with $l(t)$ the longest arc of $|z|=t$ inside $E(c)$, and the Cauchy--Bunyakovsky inequality on dyadic blocks. The statements are those the paper prints, as recorded on the source card goldberg_1979_sets_which_modulus_entire_function_has; the proofs are not verified in this corpus.

Source. A. A. Gol'dberg, Sets on which the modulus of an entire function has a lower bound (Russian), Sibirsk. Mat. Zh. 20 (1979), no. 3, 512--518, 691; the publisher's record is the linked mathnet.ru page. The record dates the article to the year alone, and the page is named by that date.

Acceptance. Refereed: the paper appeared in Sibirskii Matematicheskii Zhurnal. Reviewed: the site's curator, T. F. Bloom, labels the problem solved, credits Gol'dberg and Camera with independent proofs of Hayman's conjecture, and credits Gol'dberg with the negative answer to the second question; Hayman and Lingham's 2018 survey of Hayman's problems (library card, Update 2.40) records Gol'dberg's independent proof of the growth bound and his example for the second part. Camera's thesis is on Camera's page. Nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki.