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

Marstrand type slicing statements in $\mathbb{Z}^{2}\subset \mathbb{R}^{2}$ are false for the counting dimension