  In this addendum we strengthen the results of math-ph/0002010 in the case of
polynomial phases. We prove that Cesaro means of the pair correlation functions
of certain integrable quantum maps on the 2-sphere at level N tend almost
always to the Poisson (uniform limit). The quantum maps are exponentials of
Hamiltonians which have the form a p(I) + b I, where I is the action, where p
is a polynomial and where a,b are two real numbers. We prove that for any such
family and for almost all a,b, the pair correlation tends to Poisson on average
in N. The results involve Weyl estimates on exponential sums and new metric
results on continued fractions. They were motivated by a comparison of the
results of math-ph/0002010 with some independent results on pair correlation of
fractional parts of polynomials by Rudnick-Sarnak.
