A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets

We prove a structural upper bound on the $p$-adic valuation of denominators of rationals belonging to a missing-digit set $K_{m,D}$, generalizing a key step in recent work of Lin, Wu, and Yang [arXiv:2603.24614] on reciprocals of factorials. For a rational $\frac{r}{Q}$ with $\gcd (Q,m)=1$ and a fixed prime $p_0\nmid m$, membership in $K_{m,D}$ forces ${\nu }_{p_0}(Q)$ to be controlled by the $p_0$-adic valuation of the multiplicative order of $m$ modulo the radical of $Q$, with explicit overhead depending only on $m$ and $D$. Because the obstruction is stated at the level of a single denominator, a pair of sequence-specific valuation estimates converts it into an effective finiteness criterion for $\{\frac{1}{a_n}:n\in \mathbb{N}\}\cap K_{m,D}$. Specializing to the case in which $Q$ is the part of $n!$ coprime to $m$ recovers the fixed-prime step in the Lin–Wu–Yang argument. As applications, we treat reciprocals of superfactorials, products of polynomial values, and products of Fibonacci numbers. We also exhibit an exponential family – products of $(m^k-1)$ – for which the full structural criterion applies but a coarser largest-prime-factor formulation does not.