I will report on a new type of sub-Liouville bound in Diophantine Geometry, which concerns limiting the number of Galois orbits of algebraic points in a linear algebraic torus that have a small canonical height and get very close to a given algebraic subvariety definable over a number field. As a consequence, extending work of D. Lind to the higher rank case, the annihilator $\mathrm{Per}_N$ of $N \cdot \mathbb{Z}^d$ under a Noetherian Z^d-action by automorphisms of a compact abelian group $X$ of a finite topological entropy $h$ has $\exp((1+o(1))hN^d)$ connected components, as $N \to \infty$. Moreover, if in addition the action has a completely positive entropy, the subgroups $\mathrm{Per}_N$ equidistribute in the Haar measure of the group X as $N \to \infty$.
By a well known construction, which I will briefly expose, this result on dynamical systems of algebraic origin yields also a proof of a conjecture by Silver and Williams in geometry, giving an asymptotics of torsion in the first integral homology of the $(\mathbb{Z}/N)^d$-covering of the exterior of a link of d components inside a homology 3-sphere.