  In 1982, Lagarias showed that solving the approximate Shortest Vector Problem
also solves the problem of finding good simultaneous Diophantine
approximations. Here we provide a deterministic, dimension-preserving reduction
in the reverse direction. It has polynomial time and space complexity, and it
is gap-preserving under the appropriate norms. We also give an alternative to
the Lagarias algorithm by first reducing his version of simultaneous
approximation to one with no explicit range in which a solution is sought.
