## Abstract

We prove that some infinite

-adically discrete sets have Diophantine definitions in large subrings of 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 for every prime of

*K*there exists a set of

*K*-primes of density arbitrarily close to 1 such that there is an infinite -adically discrete set that is Diophantine over the ring of -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 of density 1 and an infinite Diophantine subset of 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 of density 1 such that there exists a Diophantine model of ℤ over . This line of research is motivated by a question of Mazur concerning the distribution of rational points on varieties in a nonarchimedean topology and questions concerning extensions of Hilbert’s Tenth Problem to subrings of number fields.

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