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

TL;DR

The paper demonstrates that Marstrand-type slicing statements do not hold for counting dimension in certain separated subsets of ^2, providing a counterexample where the expected dimension behavior under slicing fails.

Contribution

It introduces a counterexample in ^2 showing that Marstrand-type slicing theorems do not extend to counting dimension, contrasting with mass dimension results.

Findings

Counterexample set with counting dimension 1

Slicing behavior differs from mass dimension cases

Marstrand-type theorems fail for counting dimension in this context

Abstract

We show that for $1$ separated subsets of $\R^{2}$, the natural Marstrand type 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 $E$ of the plane which has counting dimension $1$, while for a positive Lebesgue measure parameter set of tubes of width $1$, the intersection of the tube with the set $E$ has counting dimension $1$. This is in contrast to the behavior of such sets with the mass dimension where the slicing theorems hold true.