$p$-adic equidistribution and an application to $S$-units

We prove a Galois equidistribution result for torsion points in ${\mathbb{G}}_m^n$ in the $p$-adic setting for test functions of the form $\log \mathrm{\mid }F{\mathrm{\mid }}_p$ where $F$ is a nonzero polynomial with coefficients in the $p$-adic numbers. Our result includes a power saving quantitative estimate of the decay rate rate of the equidistribution. As an application we show that Ih’s Conjecture is true for a class of divisors of ${\mathbb{G}}_m^n$.