Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rectangular Shrinking Targets on Self-Similar Carpets

Demi Allen, Thomas Jordan, Benjamin Ward

TL;DR

This work extends the shrinking target problem to self-similar carpets with rectangular targets, deriving an exact expression for the Hausdorff dimension of the shrinking target set in terms of the carpet's global dimension \\gamma and the horizontal-slice dimension \\gamma(z,w)_2. The authors provide a general theorem giving dim_H\\Λ_{\\lambda,\\xi}(z,w) as a limsup of a minimised ratio that blends base-b scaling with targeted rectangle sizes, and a corollary for linear shrinking rates that recovers center-dependent dimensional behavior. The proof combines precise covering arguments with a mass-distribution construction on a symbolic space associated to the IFS, exploiting a grid structure and digit-frequency data. The results show a novel center-dependence of the shrinking target dimension and connect to earlier work on self-similar and self-affine shrinking targets, with potential implications for fractal dynamics and dimension theory on non-product spaces.

Abstract

Since the introduction of the shrinking target problem by Hill and Velani in 1995 there has been a surge of interest in the area. In this paper we consider the case where the target is a rectangle, rather than a ball, and the underlying space is a self-similar carpet. We calculate the exact Hausdorff dimension of the resulting shrinking target set. Interestingly the Hausdorff dimension depends on the centre of the target, a condition uncommon in most other shrinking target type problems. This extends a theorem of Wang and Wu [Theorem 12.1, Math. Ann. 2021].

Rectangular Shrinking Targets on Self-Similar Carpets

TL;DR

This work extends the shrinking target problem to self-similar carpets with rectangular targets, deriving an exact expression for the Hausdorff dimension of the shrinking target set in terms of the carpet's global dimension \\gamma and the horizontal-slice dimension \\gamma(z,w)_2. The authors provide a general theorem giving dim_H\\Λ_{\\lambda,\\xi}(z,w) as a limsup of a minimised ratio that blends base-b scaling with targeted rectangle sizes, and a corollary for linear shrinking rates that recovers center-dependent dimensional behavior. The proof combines precise covering arguments with a mass-distribution construction on a symbolic space associated to the IFS, exploiting a grid structure and digit-frequency data. The results show a novel center-dependence of the shrinking target dimension and connect to earlier work on self-similar and self-affine shrinking targets, with potential implications for fractal dynamics and dimension theory on non-product spaces.

Abstract

Since the introduction of the shrinking target problem by Hill and Velani in 1995 there has been a surge of interest in the area. In this paper we consider the case where the target is a rectangle, rather than a ball, and the underlying space is a self-similar carpet. We calculate the exact Hausdorff dimension of the resulting shrinking target set. Interestingly the Hausdorff dimension depends on the centre of the target, a condition uncommon in most other shrinking target type problems. This extends a theorem of Wang and Wu [Theorem 12.1, Math. Ann. 2021].

Paper Structure

This paper contains 18 sections, 12 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Our setting
  3. Statements of results
  4. A special case
  5. The general result
  6. Preliminaries
  7. Proof of Proposition \ref{['relationship between sets']}
  8. Proof of Theorem \ref{['main']}
  9. Upper bound
  10. Lower bound
  11. Outline of the strategy
  12. Construction of a measure
  13. Measure of a general ball
  14. Proof of Theorem \ref{['corollary']}
  15. Further remarks
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\xi\geq \lambda>0$, let $y=(y_{1},y_{2})\in V$ be fixed, and consider the set Suppose that the digit frequencies of $y_{2}$ exist for each digit base $3$. Then where $V_{y}$ denotes the horizontal slice through $V$ at the point $y$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Lemma 3.1: Mass Distribution Principle
  • Lemma 3.2
  • ...and 13 more