In 1982, Lagarias showed that solving the approximate shortest vector problem also
solves the problem of finding good simultaneous Diophantine approximations
(SIAM J. Comput.,
14(1):196–209, 1985)). 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.
