Galois orbits of torsion points near atoral sets

Understand

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 $\mathbb{G}_m^d$.