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 $\rm RC_{PRIMES}$.

翻译：Cai和Hemachandra曾利用迭代常数设定证明Few $\subseteq$ $\oplus$P（从而FewP $\subseteq$ $\oplus$P）。本文指出，在试图捕捉的类的非确定性模糊程度与迭代常数设定中易于寻找的“目标”的密度（更精确地说，是所需的“无间隙”性质）之间存在张力。特别地，我们证明，即使对目标的间隙大小上限要求更为宽松，仍能捕捉到受模糊性限制的类。通过基于元定理的灵活方法，我们针对广泛类别实现了这一点，包括Valiant的非确定性无歧义类UP的对数模糊版本。我们的工作降低了关于无限、P-可打印素数集存在的进展所需的要求，从而证明基于素数的受限计数类有能力接受UP的超常数模糊类比。作为本工作的应用，我们证明Lenstra-Pomerance-Wagstaff猜想蕴含所有(O(1) + loglogn)模糊NP集均属于受限计数类$\rm RC_{PRIMES}$。