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, . . . , δt whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S1, . . . , St of densities δ1, . . . , δt, 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 OK,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.
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