  We establish an equidistribution result for push-forwards of certain locally
finite algebraic measures in the adelic extension of the space of lattices in
the plane. As an application of our analysis we obtain new results regarding
the asymptotic normality of the continued fraction expansions of most rationals
with a high denominator as well as an estimate on the length of their continued
fraction expansions. By similar methods we also establish a complementary
result to Zaremba's conjecture. Namely, we show that given a bound M, for any
large q, the number of rationals $p/q\in [0,1]$ for which the coefficients of
the continued fraction expansion of p/q are bounded by M is $o(q^{1-\epsilon})$
for some $\epsilon>0$ which depends on M.
