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A note on limsup sets of annuli

Mumtaz Hussain, Benjamin Ward

TL;DR

This work derives Jarník-Besicovitch-type dimensions for limsup sets formed by annuli around rational points in [0,1]^n under various norms, including standard, Euclidean, and rectangular (weighted) annuli. The authors develop and combine shifted mass transference principles for rectangles with Cassel-style scaling lemmas to obtain precise Hausdorff-dimension formulas for multiple annulus configurations, and they extend these results to perturbed inhomogeneous settings W_n^gamma(psi). A key finding is that, under slow outer-radius decay and fast inner-radius decay, the dimension can approach n-1; perturbations of centers often preserve dimension and f-measures, demonstrating robustness of the Diophantine-approximation dimension phenomena across a broad class of quasi-annular sets.

Abstract

We consider the set of points in infinitely many max-norm annuli centred at rational points in $\mathbb R^{n}$. We give Jarník-Besicovitch type theorems for this set in terms of Hausdorff dimension. Interestingly, we find that if the outer radii are decreasing sufficiently slowly, dependent only on the dimension $n$, and the thickness of the annuli is decreasing rapidly then the dimension of the set tends towards $n-1$. We also consider various other forms of annuli including rectangular annuli and quasi-annuli described by the difference between balls of two different norms. Our results are deduced through a novel combination of a version of Cassel's Scaling Lemma and a generalisation of the Mass Transference Principle, namely the Mass transference principle from rectangles to rectangles due to Wang and Wu (Math. Ann. 2021).

A note on limsup sets of annuli

TL;DR

This work derives Jarník-Besicovitch-type dimensions for limsup sets formed by annuli around rational points in [0,1]^n under various norms, including standard, Euclidean, and rectangular (weighted) annuli. The authors develop and combine shifted mass transference principles for rectangles with Cassel-style scaling lemmas to obtain precise Hausdorff-dimension formulas for multiple annulus configurations, and they extend these results to perturbed inhomogeneous settings W_n^gamma(psi). A key finding is that, under slow outer-radius decay and fast inner-radius decay, the dimension can approach n-1; perturbations of centers often preserve dimension and f-measures, demonstrating robustness of the Diophantine-approximation dimension phenomena across a broad class of quasi-annular sets.

Abstract

We consider the set of points in infinitely many max-norm annuli centred at rational points in . We give Jarník-Besicovitch type theorems for this set in terms of Hausdorff dimension. Interestingly, we find that if the outer radii are decreasing sufficiently slowly, dependent only on the dimension , and the thickness of the annuli is decreasing rapidly then the dimension of the set tends towards . We also consider various other forms of annuli including rectangular annuli and quasi-annuli described by the difference between balls of two different norms. Our results are deduced through a novel combination of a version of Cassel's Scaling Lemma and a generalisation of the Mass Transference Principle, namely the Mass transference principle from rectangles to rectangles due to Wang and Wu (Math. Ann. 2021).

Paper Structure

This paper contains 7 sections, 12 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Annuli defined by different norms
  3. Approximation by rectangular annuli
  4. Perturbed Diophantine approximation
  5. Preliminaries
  6. A shifted mass transference principle for rectangles
  7. Proof of Theorem \ref{['dim >2 generalised']}

Key Result

Theorem 1.1

Take $n\in {\mathbb{N}}$ and let $\psi(q)=q^{-\tau_{\psi}}$ and $\phi(q)=q^{-\tau_{\phi}}$ for $\tau_{\psi},\tau_{\phi}\in{\mathbb{R}}_{+}$ with $\tau_{\psi}\geq \frac{1}{n}$. Then

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Theorem 1.3
  • Remark 2
  • Theorem 2.1
  • Corollary 2.1
  • proof : Proof of Theorem \ref{['eucildean norm']}
  • Theorem 3.1: BV06
  • Lemma 3.1: BDGW23
  • ...and 11 more