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Using Poonen's version of the "weak vertical method" we produce new examples of "large" and "small" rings of algebraic numbers (including rings of integers) where Z and/or the ring of integers of a subfield are exis- tentially definable and/or where the ring version of Mazur's conjecture on the topology of rational points does not hold.

Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers