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(loglogn)-ambiguity NP sets are in the restricted counting class $\rm RC_{PRIMES}$.

翻译：Cai 和 Hemachandra 使用迭代常数设定来证明,在迭代常数设定中使用的易选“目标”的密度(或更精确地说,所需的“非突变”性)之间,存在一种矛盾。特别是,我们表明,在目标上限限制的距离上方范围更小,就能够捕获含混度的类别。我们通过灵活、基于超时代理论的方法,在一系列广泛的类别中这样做,包括Valiant 明确无误的非定义类的对数模糊性版本。我们的工作降低了在存在无限、P-可打印的百分数中取得进展的门槛。我们的工作足以表明,基于主数的有限计数类别具有接受超常识性美元类别。我们接受超常代言调类类类的模拟性能,意味着我们正统的常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任常任。