  We prove that the Galois equidistribution of torsion points of the algebraic
torus $\mathbb{G}_{m}^d$ extends to the singular test functions of the form
$\log{|P|}$, where $P$ is a Laurent polynomial having algebraic coefficients
that vanishes on the unit real $d$-torus in a set whose Zariski closure in
$\mathbb{G}_m^d$ has codimension at least $2$. Our result includes a power
saving quantitative estimate of the decay rate of the equidistribution. It
refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also
supplies a purely Diophantine proof. As an application, we confirm Ih's
integrality finiteness conjecture on torsion points for a class of atoral
divisors of $\mathbb{G}_m^d$.
