Galois orbits of torsion points near atoral sets

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 $\log <!--FUNCTION\; APPLICATION-->\left|P\right|$, 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 ${\mathbb{𝔾}}_m^d$.