The growth of entire functions of exponential type and characteristics of distributions of points along a straight line

Abstract: Let $Z$ and $W$ be a pair of point distributions of finite upper density on the complex plane $\mathbb C$ with the real axis $\mathbb R$. We give several variants of necessary and at the same time sufficient conditions for their arrangement, under which for every entire function of exponential type $g\neq 0$ vanishing on $W$, there is, respectively, either an entire function of exponential type $f\neq 0$ vanishing on $Z$ and satisfies one of the two variants of the constraints: 1) $|f(iy)|\leq |g(iy)|$ at each $y\in \mathbb R$, i.e. everywhere on the imaginary axis $i\mathbb R$, 2) $\ln |f(iy)|\leq \ln |g(iy)|+o(|y|)$ as $y\to \pm \infty$, or for any number $\varepsilon >0$ there is an entire function of exponential type $f\neq 0$ vanishing on $Z$ and satisfies the inequality $\ln \bigl|f(iy)\bigr|\leq \ln \bigl|g(iy)\bigr|+\varepsilon |y|$ for all $y\in \mathbb R\setminus E$, where $E\subset \mathbb R$ is a set of finite linear Lebesgue measure. Our study is carried out within the framework of generalization of the development of the classical theorem of P. Malliavin and L.A. Rubel of the 1960s, where the case of the location of $Z\subset \mathbb R^+$ and $W\subset \mathbb R^+$ on the positive semiaxis $\mathbb R^+\subset \mathbb R$ is considered. Our criteria are given in terms of special logarithmic characteristics and (sub)measures for $Z$ and $W$. At the same time, in the third variant, there are no additional requirements for $Z$ and $W$, but in the first and second variants, we require the asymptotic separation of $Z$ and $W$ by angles from the imaginary axis together with the Lindel\"of-type condition for $W$ along the imaginary axis $iR$ on a certain symmetry of the imaginary parts of $1/w$ for $w\in W$ in the form $\Bigl|\sum_{1\leq |w|\leq r}\Im (1/w)\Bigr|=O(1)$ as $r\to +\infty$.