  We show that for 1 separated subsets of $\mathbb{R}^{2}$, the natural
Marstrand like slicing statements are false with the counting dimension that
was used earlier by Moreira and Lima and variants of which were introduced
earlier in different contexts. We construct a 1 separated subset of the plane
which has counting dimension one while a positive Lebesgue measure of width 1
tubes also all have counting dimension one. This is in contrast to the behavior
of such sets with the mass dimension where the slicing theorems holds true.
