Distributions of (non)uniqueness for entire functions of arbitrary growth

A simple uniqueness theorem is given for entire functions $f$ on the complex plane $\mathbb{C}$ with upper constraints on the growth of its module $\ln|f|\leq M$. The result is formulated exclusively in terms of the radial integral counting function $\mathsf{N}_Z$ of the distribution of points $Z$, such that $f(Z)=0$. In the opposite direction, a rather general nonuniqueness theorem is obtained on the existence of a nonzero entire function $f$ that vanishes on $Z$, with restrictions on the growth of $\ln|f|$ by small shifts of the countable function $\mathsf{N}_Z$.