Taylor coefficients and zeroes of entire functions of exponential type
Lior Hadassi, Mikhail Sodin
TL;DR
The paper investigates how a sequence of multipliers $\omega_n$ governs the zero distribution of $F(z)=\sum_{n\ge 0}\omega_n\frac{z^n}{n!}$, under the umbrella of entire functions of exponential type. It develops a novel self-correlation framework and analytic interpolation to relate local unimodular structure of the coefficients to global zero growth, establishing a density-driven dichotomy: either the zero-count $n_F(R)$ grows linearly or $F$ is an exponential function. A second main result shows that if the coefficients are bounded away from zero and infinity and $n_F(R)=o(\sqrt{R})$ (or along a subsequence $n_F(r_j)=O(r_j^{\alpha})$ with $\alpha<1/2$), then $F$ must be exponential; this is complemented by a sharp example illustrating the bound is tight. The paper also presents sharpness and subharmonic counterexamples to delineate the limits of the methods, highlighting the delicate balance between zero distribution and coefficient structure in EFETs.
Abstract
Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} ω_n \frac{z^n}{n!} \] with unimodular coefficients $|ω_n|=1$. We show that either the counting function $n_F(r)$ of zeroes of $F$ grows linearly at infinity, or $F$ is an exponential function. The same conclusion holds if only a positive asymptotic proportion of the coefficients $ω_n$ is unimodular. This significantly extends a classical result of Carlson (1915). The second result requires less from the coefficient sequence $ω$, but more from the counting function of zeroes $n_F$. Assuming that $0<c\le |ω_n| \le C <\infty$, $n\in\mathbb Z_+$, we show that $n_F(r) = o(\sqrt{r})$ as $r\to\infty$, implies that $F$ is an exponential function. The same conclusion holds if, for some $α<1/2$, $n_F(r_j)=O(r_j^α)$ only along a sequence $r_j\to\infty$. Furthermore, this conclusion ceases to hold if $n_F(r)=O(\sqrt r)$ as $r\to\infty$.