---

Statement

Setting (p. 62). As in Theorem 1: $f(z)=\sum_{n\ge0}a_nz^{\lambda_n}$ is entire, (1), with $\lambda_n$ a strictly increasing sequence of non-negative integers, and $M(r)$, $m(r)$ and $\mu(r)$ are its maximum modulus, minimum modulus and maximum term.

Theorem 3 (p. 63). If for a positive integer $h$

$$\sum_{n=0}^\infty\frac{1}{\lambda_{n+h}-\lambda_n}<\infty,\qquad(7)$$

then

$$\limsup_{r\to\infty}\frac{\mu(r)}{M(r)}\ge\frac{1}{2h-1};\qquad(8)$$

but if

$$\sum_{n=0}^\infty\frac{1}{\lambda_{n+h}-\lambda_n}=\infty\qquad(9)$$

for every $h$, then there is an entire function of the form (1) such that

$$\lim_{r\to\infty}\frac{\mu(r)}{M(r)}=\lim_{r\to\infty}\frac{m(r)}{M(r)}=0.\qquad(10)$$

For $h=1$, (7) is the hypothesis (4) of Theorem 1. The paper adds (p. 63) that the conjecture that (7) gives $\limsup m(r)/M(r)>0$, (11), is disproved by the function

$$\sum_{n=0}^\infty\frac{z^{n^3}}{(n^3)!}+\sum_{n=0}^\infty\frac{z^{n^3+1}}{(n^3+1)!}.$$

Proof pointer

Pp. 66--68, Section 4. For (8), with $h>1$, the block averages of Theorem 1's proof are formed from $\epsilon_n=(\lambda_{n+h}-\lambda_n)^{-1}$; at suitable radii the terms more than $h-1$ places from the maximum term sum to $o(\mu(r))$, (36), while the at most $2h-2$ nearer terms are each at most $\mu(r)$, (35), which gives $\liminf M(r)/\mu(r)\le2h-1$. For (10), one of the $h$ series (37) along a residue class of indices diverges, a function built on that subsequence as in Theorem 2 is spread over blocks of $h$ nearly equal terms, (40)--(41), and $M(r)>(h+1-\epsilon)\mu(r)$ and $m(r)\le(h+1-\epsilon)^{-1/2}M(r)$ follow, (42)--(45). The paper says (p. 68) that this does not quite complete the proof, since these bounds, though arbitrarily small, are not zero, and that a subsequence $\lambda_n^*$ whose intervals contain an increasing number of the $\lambda_n$, with $\sum(\lambda_{n+1}^*-\lambda_n^*)^{-1}$ still divergent, would finish it; it does not give those details.

Read depth

Claims checked: Theorem 3, (7) to (11) and the example were read clause by clause on the page images of the print, and the proof on pp. 66--68 was followed for structure. The last step of the proof of the second part is only indicated in the paper. Nothing here is independently reviewed.

Dependencies

The proof reuses the inequality (14)--(15) from the proof of Theorem 1 and the construction of Theorem 2.

Source. P. Erdős and A. J. Macintyre, Integral functions with gap power series, Proc. Edinburgh Math. Soc. (2) 10 (1954), 62--70; the edition read is named on the source card.

Bears on

None among the problem pages. The theorem bounds the maximum term against the maximum modulus and, in its second part and the example, shows that $m(r)/M(r)$ can tend to zero; it states nothing about $\log m(r)/\log M(r)$, the quantity of Problem 516.