Hilbert's tenth problem and Mazur's conjectures in complementary subrings of number fields

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 δ₁, . . . , δ_t whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S₁, . . . , S_t of densities δ₁, . . . , δ_t, respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring O _K,Si. We also show that we can find a partition as above such that each ring O_K,Si 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.