A finite-order answer to a problem of Erdős on maximum modulus points

In 1964, Erd\H{o}s asked whether, for a non-monomial entire function, the number of maximum modulus points on the circle $|z|=r$ can become arbitrarily large as $r\to\infty$. In 1968, Herzog and Piranian answered this question affirmatively, but without quantitative control on the resulting function. We prove that such an example can be chosen to have finite order and, moreover, to belong to the Eremenko--Lyubich class $\B$. We also prove an interpolation theorem for maximum modulus sets: prescribed points with pairwise distinct moduli can be forced to lie in the maximum modulus set of some function in $\B$, with finite order under a geometric separation condition.