Convergence to the Mahler measure and the distribution of periodic points for algebraic Noetherian ${\mathbb{Z}}^d$-actions

For every $P \in $\mathbb{Z}$[x_1^{\pm 1}, \ldots, x_d^{\pm 1}] \setminus {0}$,andevery$\varepsilon > 0$, we prove that there are a computable function $M=M(d,\epsilon ,\mathrm{deg}P,h(P))<\mathrm{\infty }$ and a finite union $Z = Z(d,\varepsilon,\deg{P},h(P))$ofpropertorsioncosets$\boldsymbol{\mu} T \subsetneq $\mathbb{G}$_m^d$suchthat,forevery$N \in $\mathbb{N}$$,$Z$containsallbutatmost$M$ofthetorsionpoints$\boldsymbol{\zeta} \in \mu_N^d$satisfying$|P(\boldsymbol{\zeta})| < e^{-\varepsilon \phi(N)}$. This extends a well known structural theorem from torsion points lying exactly on a variety to torsion points lying very near to the subvariety. As a consequence, we prove that the averages of $\log \mathrm{\mid }P(\mathbf{x})\mathrm{\mid }$ over ${\mu }_N^d$ converge as $N\to \mathrm{\infty }$ to the Mahler measure of $P$. By the work of B. Kitchens, D. Lind, K. Schmidt and T. Ward, the convergence consequence amounts to the following statement in dynamics: For every Noetherian ${\mathbb{Z}}^d$-action $T:{\mathbb{Z}}^d\to \mathrm{A}\mathrm{u}\mathrm{t}(X)$ by automorphisms of a compact abelian group $X$ having a finite topological entropy $h(T)$, the exponential growth rate of the number of connected components of the group ${\mathrm{P}\mathrm{e}\mathrm{r}}_N(T)$ of $N\cdot {\mathbb{Z}}^d$-periodic points of $(X,T)$ exists as $N\to \mathrm{\infty }$, and equals the topological entropy $h(T)$. Moreover, it follows that all weak-$\ast$ limit measures of the push-forwards of the Haar measures on ${\mathrm{P}\mathrm{e}\mathrm{r}}_N(T)$, under any a sequence of positive integers $N$, are measures of maximum entropy $h(T)$. Our main arithmetic result extends to Diophantine approximation by points of sufficiently small canonical height. It is best possible in such a generality, where an exceptional set is an inevitable feature.