  Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers is one
of the biggest open problems in the area of undecidability in number theory. In
this paper we construct new, computably presentable subrings $R$ of $\mathbb Q$
having the property that Hilbert's Tenth Problem for $R$, denoted $HTP(R)$, is
Turing equivalent to $HTP(\mathbb Q)$.
  We are able to put several additional constraints on the rings $R$ that we
construct. Given any computable nonnegative real number $r \leq 1$ we construct
such a ring $R = Z[\frac1p : p \in S]$ with $S$ a set of primes of lower
density $r$. We also construct examples of rings $R$ for which deciding
membership in $R$ is Turing equivalent to deciding $HTP(R)$ and also equivalent
to deciding $HTP(\mathbb Q)$. Alternatively, we can make $HTP(R)$ have
arbitrary computably enumerable degree above $HTP(\mathbb Q)$. Finally, we show
that the same can be done for subrings of number fields and their prime ideals.
