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

Aritro Pathak

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.

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

Abstract

We show that for separated subsets of , 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 separated subset of the plane which has counting dimension , while for a positive Lebesgue measure parameter set of tubes of width , the intersection of the tube with the set has counting dimension . This is in contrast to the behavior of such sets with the mass dimension where the slicing theorems hold true.

Paper Structure

This paper contains 4 sections, 6 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction and statement of results.
  2. Slicing results with the counting dimension.
  3. Further questions.
  4. Acknowledgements:

Key Result

Theorem 1

Let $E \subseteq \mathbb{R}^2$ be a $1$ separated set. Then for all $v \in \mathbb{R}$, for Lebesgue-a.e. $u \in \mathbb{R}_+$,

Figures (2)

  • Figure 1: The tube $t_{u,0}$, with $v=0$ is shown, and the perpendicular line $t_{\perp}$. A tube $t_{u,v}$ with $v\neq 0$, is a parallel translation of $t_{u,0}$, along the line $t_{\perp}$.
  • Figure 2: The cone showing the clusters at any specific level of the construction of the set $E$, growing 'diagonally' from one edge of the cone to the other. Every level of $E$ consists of such a diagonal sequence of points, and only one level of $E$ is shown in the figure. The width of each cluster is the same $w$, while the height of each cluster at the level $k$, is given by the sequence $n_k \to\infty$ as $k\to \infty$, which for simplicity can be chosen to be $n_k =k, \ \forall k\in \mathbb{Z}$.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof : Proof of \ref{['thm:strongerslicing1']}, (2)
  • proof : Proof of \ref{['thm:strongerslicing2']}
  • proof : Proof of \ref{['thm:strongslicing']}
  • proof : Proof of \ref{['thm:weakreal']}