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We give the first examples of infinite sets of primes S such that Hilbert'sTenth Problem over Z[S^{-1}] has a negative answer. In fact, we can take S tobe a density 1 set of primes. We show also that for some such S there is apunctured elliptic curve E' over Z[S^{-1}] such that the topological closure ofE'(Z[S^{-1}]) in E'(R) has infinitely many connected components.Comment: 10 page

Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of $\mathbb {Q}$