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 ${\mathbb{𝔾}}_m^d$.

中文翻译：

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接近 atoral sets 的扭转点的伽罗瓦轨道

我们证明代数环面 ${\mathbb{𝔾}}_m^d$ 的扭点的伽罗瓦等 $d$ 分分布扩展到形式为 $\log \left|P\right|$ 的奇异测试函数，其中 2 的共维至少为 2。我们的结果包括对等分布衰减率的节能定量估计。它提炼了 Lind、Schmidt 和 Verbitskiy 的遍历定理，其中还提供了一个纯粹的丢番图证明。作为一个应用，我们确认了 的一类对数约数的扭转点的 Ih 积分有限性猜想 ${\mathbb{𝔾}}_m^d$ 。