  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.
