  We show that Hilbert's Tenth Problem is undecidable for complementary
subrings of number fields and that the p-adic and archimedean ring versions of
Mazur's conjectures do not hold in these rings. More specifically, given a
number field K, a positive integer t>1, and t nonnegative computable real
numbers delta_1,..., delta_t whose sum is one, we prove that the nonarchimedean
primes of K can be partitioned into t disjoint recursive subsets S_1,..., S_t
of densities delta_1,..., delta_t, respectively such that Hilbert's Tenth
Problem is undecidable for each corresponding ring O_{K,S_i}. We also show that
we can find a partition as above such that each ring O_{K,S_i} possesses an
infinite Diophantine set which is discrete in every topology of the field. The
only assumption on K we need is that there is an elliptic curve of rank one
defined over K.
