This dissertation addresses the problem of numerical integration of multivariate functions with logarithmic singularities. For all non-zero integer Laurent polynomials P in d commuting variables, we prove that the averages of log |P(x)| over ζ εμ[special characters omitted], P(ζ) ≠ 0, converge as N → ∞ to the Mahler measure of P. To this end we prove a sub-Liouville bound, up to a uniformly bounded exceptional set that varies with a number field K, on the distance from a small point of [special characters omitted][special characters omitted] (K) to a fixed algebraic subset of [special characters omitted][special characters omitted]under a fixed Archimedean place, [special characters omitted] →[special characters omitted].

By the work of B. Kitchens. D. Lind; K. Schmidt and T. Ward, the convergence over the finite groups μ[special characters omitted] amounts to the following statement in dynamics: For every Noetherian [special characters omitted]^d-action T : [special characters omitted]^d → Aut(X) by automorphisms of a compact abelian group X having a finite topological entropy h(T), the group Per_N( T) ⊂ X of N · [special characters omitted]^d-periodic points of T has e^{(1+o(1)) h(T ) Nd} connected components, as N → ∞. Moreover, it follows that all weak-* limit measures of the push-forwards of the Haar measures on Per _N(T), under any a sequence of positive integers N, are measures of maximum entropy h(T). Combined with log work of Thang Le, this also completes the solution of the abelian case of the problem of the asymptotic growth of torsion in the homology of congruence covers of a fixed finite simplicial complex.

An alternative and entirely different route to the convergence result and its consequences was attained recently by Habegger in his work on Diophantine approximation to definable sets. We discuss the comparison of the two results and some further problems that they raise.