  We prove that most one-dimensional projections of a discrete subset of a
plane are either dense in R (the real line), or form a discrete subset of R.
More precisely, the set E of exceptional directions (for which the indicated
dichotomy fails) is a meager subset of the unit circle T of Lebesgue measure 0.
The set E however does not need to be small in the sense of Hausdorff
dimension.
