A note on limsup sets of annuli

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).