  We consider the problem of digitalizing Euclidean segments. Specifically, we
look for a constructive method to connect any two points in $\mathbb{Z}^d$. The
construction must be {\em consistent} (that is, satisfy the natural extension
of the Euclidean axioms) while resembling them as much as possible. Previous
work has shown asymptotically tight results in two dimensions with $\Theta(\log
N)$ error, where resemblance between segments is measured with the Hausdorff
distance, and $N$ is the $L_1$ distance between the two points. This
construction was considered tight because of a $\Omega(\log N)$ lower bound
that applies to any consistent construction in $\mathbb{Z}^2$.
  In this paper we observe that the lower bound does not directly extend to
higher dimensions. We give an alternative argument showing that any consistent
construction in $d$ dimensions must have $\Omega(\log^{1/(d-1)} N)$ error. We
tie the error of a consistent construction in high dimensions to the error of
similar {\em weak} constructions in two dimensions (constructions for which
some points need not satisfy all the axioms). This not only opens the
possibility for having constructions with $o(\log N)$ error in high dimensions,
but also opens up an interesting line of research in the tradeoff between the
number of axiom violations and the error of the construction. In order to show
our lower bound, we also consider a colored variation of the concept of
discrepancy of a set of points that we find of independent interest.
