  Cai and Hemachandra used iterative constant-setting to prove that Few
$\subseteq$ $\oplus$P (and thus that FewP $\subseteq$ $\oplus$P). In this
paper, we note that there is a tension between the nondeterministic ambiguity
of the class one is seeking to capture, and the density (or, to be more
precise, the needed "nongappy"-ness) of the easy-to-find "targets" used in
iterative constant-setting. In particular, we show that even less restrictive
gap-size upper bounds regarding the targets allow one to capture
ambiguity-limited classes. Through a flexible, metatheorem-based approach, we
do so for a wide range of classes including the logarithmic-ambiguity version
of Valiant's unambiguous nondeterminism class UP. Our work lowers the bar for
what advances regarding the existence of infinite, P-printable sets of primes
would suffice to show that restricted counting classes based on the primes have
the power to accept superconstant-ambiguity analogues of UP. As an application
of our work, we prove that the Lenstra-Pomerance-Wagstaff Conjecture implies
that all (O(1) + loglogn)-ambiguity NP sets are in the restricted counting
class $\rm RC_{PRIMES}$.
