A note on limsup sets of annuli

TL;DR

This paper investigates the Hausdorff dimension of points lying in infinitely many annuli centered at rational points, revealing how the dimension varies with the annuli's radii and thickness, using advanced measure theory techniques.

Contribution

It introduces new Jarník-Besicovitch type theorems for limsup sets of annuli and extends the mass transference principle to more general geometric shapes.

Findings

Dimension tends to n-1 when radii decrease slowly and thickness decreases rapidly.

Results apply to various annuli including rectangular and quasi-annuli.

Novel combination of scaling lemma and generalized mass transference principle.

Abstract

We consider the set of points in infinitely many max-norm annuli centred at rational points in $\mathbb R^{n}$. We give Jarn\'ik-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).