Galois orbits of torsion points near atoral sets

We prove that the Galois equidistribution of torsion points of the algebraic torus Gₘᵈ 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 Gₘᵈ 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 Gₘᵈ.