Given an invertible affine transformation $f:\Bbb R^n\to\Bbb R^n$ represented as $f(x)=Ax+b$, I'd like to know whether $f((0,1)^n)\cap\Bbb Z^n$ is nonempty. Is there an efficient way to compute this?
My current "approximate" approach is to evaluate $f^{-1}$ at a few integer points near $f(\frac12,\frac12,\dots)$, but naively, to make this correct, I think I'd need to search a region containing exponentially many (in $n$) integer points.
(Related, unanswered question: Finding all lattice points in an $n$-dimensional hypercube)