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Claim. Let $f(z)=\sum_{k\ge1}a_kz^{n_k}$ be an entire function with $a_k\ne0$ for every $k$ and $\sum_k1/n_k<\infty$ (Fejér gaps). Takafumi Murai, The deficiency of entire functions with Fejér gaps, Ann. Inst. Fourier 33 (1983), no. 3, 39--58, doi:10.5802/aif.930, proves that $f$ has no finite deficient value in the sense of Nevanlinna theory: $\delta(a,f)=0$ for every $a\in\mathbb C$. The theorem is recorded on its library card, which also records the paper's engine, a proximity-function estimate $m(r,f)>(1-\epsilon)\log M(r,f)$ outside a set of finite logarithmic measure. The consequence for the question is immediate: $f$ is transcendental, so $T(r,f)/\log r\to\infty$, while a value $a$ taken only finitely often has $N(r,a)=O(\log r)$ and hence $\delta(a,f)=1$; so every value is taken infinitely often. The paper presents its theorem as a strengthening of this classical Fejér--Biernacki theorem, the result of Biernacki 1927. Section 5 of the paper shows that the gap hypothesis cannot be weakened to Fabry gaps, $k/n_k\to0$: it constructs an entire function with Fabry gaps whose deficiency at $0$ equals $1$. That example does not bear on the question, since deficiency $1$ at $0$ does not show that $0$ is taken finitely often. The paper was cited on the site's discussion thread on 2026-03-26 as related to the problem.

Covers. Every instance of Problem 517 with $\sum_k1/n_k<\infty$, of any order; such exponents satisfy $n_k/k\to\infty$, as the Biernacki page shows. Functions with $n_k/k\to\infty$ and $\sum1/n_k=\infty$ are not covered; the finite-order ones are settled by Pólya 1929, and the infinite-order ones remain open.

Depends on. Nothing in this wiki: the theorem is the paper's own, and the deduction of infinitely many $a$-points from zero deficiency is the elementary Nevanlinna-theory step above.

Acceptance. Refereed: a journal publication, Annales de l'Institut Fourier 33 (1983), no. 3, the DOI linked above; the publisher's record gives only the year, so the page is dated to its first day. Reviewed is not listed: the site labels the problem OPEN, its commentary does not mention the paper, and a thread comment is not acceptance. Formalized is not listed: no Lean statement or proof of the theorem is recorded.