A simple uniqueness theorem is given for entire functions f on the complex plane \(\mathbb {C}\) with upper constraints on the growth of its modulus \(\ln |f|\le M\) . The result is formulated exclusively in terms of the radial integral counting function \(\textsf{N}_Z\) 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 \(\textsf{N}_Z\) .
