We prove that infinite p-adically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree-2 extension of a totally real number field, then there exists a prime p of K and a set of K-primes S of density arbitrarily close to 1 such that there is an infinite p-adically discrete set that is Diophantine over the ring O K,S of S-integers in K. Second, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes S of density 1 and an infinite Diophantine subset of O K,S that is v-adically discrete for every place v of K. Third, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes S of density 1 such that there exists a Diophantine model of Z over O K,S . This line of research is motivated by a question of Mazur concerning the distribution of rational points on varieties in a non-archimedean topology and questions concerning extensions of Hilbert's Tenth Problem to subrings of number fields.

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We construct a fully faithful functor from the category of graphs to the category of fields. Using this functor, we resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S, there exists a countable field F with the same essential computable-model-theoretic properties as S. Along the way, we develop a new "computable category theory", and prove that our functor and its partiallydefined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.

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We extend the solution of Hilbert's tenth problem to algebraic number fields having one pair of complex conjugated embeddings. The proof is based on the extended method of J. Denef used for totally real algebraic number fields. This paper extends a solution to Hilbert's tenth problem to algebraic number fields having one pair of complex conjugate embeddings. ' In one of its formulations Hilbert's tenth problem can be stated as follows. 'A similar result has recently been attained by