Hausdorff dimension of limsup sets of random rectangles in products of regular spaces

Ekström, Fredrik; Järvenpää, Esa; Järvenpää, Maarit; Суомала, Вилле

doi:10.1090/proc/13920

Cited by 10 publications

(6 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This theorem implies the main theorems in [12]. Further, a straightforward modification of our method would also yield versions of the the results of [17] for open sets.…”

Section: 2mentioning

confidence: 59%

A New Hausdorff Content Bound for Limsup Sets

Eriksson-Bique¹

2022

Preprint

View full text Add to dashboard Cite

We give a new Hausdorff content bound for limsup sets, which is related to Falconer's sets of large intersection. Falconer's sets of large intersection satisfy a content bound for all balls in a space. In comparison, our main theorem only assumes a scale-invariant bound for the balls forming the limit superior set in question.We give four applications of these ideas and our main theorem: a new proof and generalization of the mass transference principle related to Diophantine approximations, a related result on random limsup sets, a new proof of Federer's characterization of sets of finite perimeter and a statement concerning generic paths and the measure theoretic boundary. The new general mass transference principle transfers a content bound of one collection of balls, to the content bound of another collection of sets -however, this content bound must hold on all balls in the space. The benefit of our approach is greatly simplified arguments as well as new tools to estimate Hausdorff content.The new methods allow for us to dispense with many of the assumptions in prior work. Specifically, our general Mass Transference Principle, and bounds on random limsup sets, do not assume Ahlfors regularity. Further, they apply to any complete metric space. This generality is made possible by the fact that our general Hausdorff content estimate applies to limsup sets in any complete metric space.

show abstract

“…This theorem implies the main theorems in [12]. Further, a straightforward modification of our method would also yield versions of the the results of [17] for open sets.…”

Section: 2mentioning

confidence: 59%

A New Hausdorff Content Bound for Limsup Sets

Eriksson-Bique¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…For a survey on mass transference principles, one is also referred to Allen & Troscheit [3]. For dimensional results of random limsup sets, one is refer to [22,23,26,27,40] and references therein.…”

Section: 2mentioning

confidence: 99%

Mass transference principle from rectangles to rectangles in Diophantine approximation

Wang

2019

Preprint

View full text Add to dashboard Cite

By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles, i.e., if a sequence of rectangles forms a ubiquity system (a full measure property), then the limsup set defined by shrinking these rectangles to smaller rectangles has full Hausdorff measure or to say transfer a full measure property to a full Hausdorff measure property for brevity.The limsup sets generated by balls or generated by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet's theorem, the other follows from Minkowski's theorem. So the result sets up a general principle for the Hausdorff measure theory for high dimensional Diophantine approximation which, together with the landmark work of Beresnevich & Velani in 2006 where a transference principle from balls to balls is established, gives a coherent Hausdorff measure theory for metric Diophantine approximation.The dimensional theory for limsup sets generated by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods or even their generalizations fail to work.

show abstract

“…In [12], the authors consider random limsup sets generated by rectangles in products of Ahlfors regular metric spaces. There, the Lipschitz continuity of projections plays an essential role, hence the same methods are not readily available when considering rectangles in Heisenberg groups and some new machinery is required.…”

Section: Introductionmentioning

confidence: 99%