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Statement

Setting (pp. 39--41). For an entire function $g$ with $S(g)=\{n\ge1;\ c_n\ne0\}=(n_k)_{k\ge1}$ listed increasingly, $g$ has Fabry gaps if $\lim_{k\to\infty}k/n_k=0$ (p. 39). The deficiency is $\delta(a,g)=1-\limsup_{r\to\infty}N(r,a,g)/m(r,g)$, with $m(r,g)$ the characteristic function (pp. 40--41).

Construction (Section 5, heading p. 52). There is an entire function $g_\infty$ with Fabry gaps such that $\delta(0,g_\infty)=1$ (stated pp. 52 and 55). The section is unnumbered as a result; its heading reads "An entire function with Fabry gaps such that $\delta(0,\cdot)=1$" and it concludes that "the assertion of our theorem does not hold with Fejér gaps replaced by Fabry gaps" (p. 55).

The section proves only the deficiency statement. It bounds the zeros of $g_\infty$ in growing disks, $n(r,0,g_\infty)\le d_m$ for $r_m<r\le r_{m+1}$ (p. 55), and does not show that $0$ is taken only finitely often.

Source. Section 5, pp. 52--55, of Takafumi Murai, The deficiency of entire functions with Fejér gaps, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 39--58, doi:10.5802/aif.930, as identified on the source card.

Read depth. Claims checked: the statement and the definitions it uses were read on the printed pages. The construction (pp. 52--55) was read for its mechanism and not checked step by step; nothing here is independently reviewed.

Proof pointer

Section 5 (pp. 52--55). The model is $e^{z^p}$, which omits $0$ and has exponents $\{pn;\ n\ge1\}$. Write $pr(g,d)$ for the Taylor polynomial of $g$ of degree $d$. Starting from $h_0(z)=e^z$, $r_0=1$, $r_0'=2$, $q_0=1$, the paper chooses inductively a degree $d_m$ and sets $g_m=pr(h_{m-1},d_m)$, close to $h_{m-1}$ on a disk $D_{r_m}$ with the same zeros there (28)--(30); then a large integer $q_m$ and $h_m(z)=g_m(z)\exp(z/r_{m-1}')^{q_m}$, with $d_m/q_m\le2^{-m}/10$ (31) and $h_m$ large on many short arcs of every circle $|z|=r\ge r_m$ (33) (p. 53). The limit $g_\infty=\lim g_m$ is entire (p. 54). Its exponents up to $d_{m+1}$ are those of $h_m$, namely the numbers $\ell q_m+n$ with $\ell\ge0$ and $n\in S(g_m)$ and the multiples $\ell q_m$ with $\ell\ge1$ (34), which gives $\omega(r)/r\le2^{-m}$ for $d_m<r\le d_{m+1}$ and hence Fabry gaps (pp. 54--55). Rouché's theorem gives $N(r,0,g_\infty)\le d_m\log r$ (35), while the arcs of (33) give $m(r,g_\infty)\ge Cd_mr/(2\pi)-\log4$ for $r_m<r\le r_{m+1}$ (36)--(37); so $N(r,0,g_\infty)/m(r,g_\infty)\to0$ (p. 55).

Dependencies

Rouché's theorem; nothing else from the paper.

Bears on

• Problem 517: no direct bearing. A function with Fabry gaps satisfies the problem's hypothesis $n_k/k\to\infty$, but deficiency $1$ at $0$ does not mean that $0$ is taken finitely often, and the paper does not show that it is; so the example neither answers the problem nor is a counterexample to it. It shows only that the Theorem cannot be extended to Fabry gaps.