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We prove that the Galois equidistribution of torsion points of the algebraictorus $\mathbb{G}_{m}^d$ extends to the singular test functions of the form$\log{|P|}$, where $P$ is a Laurent polynomial having algebraic coefficientsthat 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 powersaving quantitative estimate of the decay rate of the equidistribution. Itrefines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it alsosupplies a purely Diophantine proof. As an application, we confirm Ih'sintegrality finiteness conjecture on torsion points for a class of atoraldivisors of $\mathbb{G}_m^d$.

Galois orbits of torsion points near atoral sets