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A Marstrand type slicing theorem for subsets of $\mathbb{Z}^2 \subset \mathbb{R}^2$ with the mass dimension

Aritro Pathak

Abstract

We prove a Marstrand type slicing theorem for the subsets of the integer square lattice. This problem is the dual of the corresponding projection theorem, which was considered by Glasscock, and Lima and Moreira, with the mass and counting dimensions applied to subsets of $\mathbb{Z}^{d}$. In this paper, more generally we deal with a subset of the plane that is $1$ separated, and the result for subsets of the integer lattice follow as a special case. We show that the natural slicing question in this setting is true with the mass dimension.

A Marstrand type slicing theorem for subsets of $\mathbb{Z}^2 \subset \mathbb{R}^2$ with the mass dimension

Abstract

We prove a Marstrand type slicing theorem for the subsets of the integer square lattice. This problem is the dual of the corresponding projection theorem, which was considered by Glasscock, and Lima and Moreira, with the mass and counting dimensions applied to subsets of . In this paper, more generally we deal with a subset of the plane that is separated, and the result for subsets of the integer lattice follow as a special case. We show that the natural slicing question in this setting is true with the mass dimension.

Paper Structure

This paper contains 9 sections, 11 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction and statement of results.
  2. Dimensions of subsets of $\mathbb{R}^{2}$ and $\mathbb{Z}^2$, and the slicing theorem.
  3. Statement of results
  4. Outline of proof of \ref{['thm:strongerslicing']}
  5. Asymptotic results for $\mathbb{F}_{p}^2$ and $\mathbb{Z}^2$
  6. Mass dimensions of specific sets and their slices.
  7. Proof of the slicing theorem.
  8. Future directions:
  9. Acknowedgements.

Key Result

Theorem \oldthetheorem

$\text{dim}_H(E\cap l)\leq \text{max}(\text{dim}_H(E)-1,0)$ for almost all straight lines $l$ in the plane.

Figures (2)

  • Figure 2: The tube $t_{(u,0)}$ and the 'box' $B_{n}(u)$ are illustrated; here $u\in O$ and the lines $l_1$ and $l_2$ are the delimiting lines of the open cone of width $\theta$ corresponding to $O$.
  • Figure 3: For a fixed $l\in \mathbb{N}$, an example of an $m_l$ 'profile' is shown with the $u$-coordinates restricted to the subset $L\cap O$, where $O$ corresponds to the cone illustrated above. The function $u \to m_{l}(u)$ for a fixed $l$ is discrete as explained in the proof, thus takes countable many values. There may be instances of the values diverging, for example to some line $l_3$ from the left as shown above, in which case by removing from the domain of the function arbitrarily small neighborhoods around these diverging points (By removing a set of tubes whose $u$ values lie in a set of measure $\epsilon/2^{l}$, the above function is bounded).

Theorems & Definitions (17)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • proof : Proof of \ref{['thm:weakfinite ']}
  • ...and 7 more