2016/11/15 by Dimitrov, Vesselin
#11G50 #11J17 #11J25 #11J87 #11K60 #11R06 #37A25 #37A35 #37A45 #57M25 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1611.04664
For every P ∈ ℤ[x1± 1, …, xd± 1] ∖ \0\, and every ε > 0, we prove that there are a computable function M = M(d,ε,°P,h(P)) < ∞ and a finite union Z = Z(d,ε,°P,h(P)) of proper torsion cosets \boldsymbolμ T \subsetneq \mathbbGmd such that, for every N ∈ ℕ, Z contains all but at most M of the torsion points \boldsymbolζ ∈ μNd satisfying |P(\boldsymbolζ)| < e-ε ϕ(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|P(x)| over μNd converge as N → ∞ 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 ℤd-action T : ℤd → Aut(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 PerN(T) of N ⋅ ℤd-periodic points of (X,T) exists as N → ∞, and equals the topological entropy h(T). Moreover, it follows that all weak-* limit measures of the push-forwards of the Haar measures on PerN(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.