Abstract
We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described within this framework. The discussed applications include the classical homogeneous and inhomogeneous approximation, the approximation by algebraic numbers, the approximation by fractional parts, the study of uniform and Poisson random coverings, and the multifractal analysis of Levy processes.
Highlights
The aim of these notes is to present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type.Such sets arise naturally in Diophantine approximation and multifractal analysis
Through the mass and the large intersection transference principles, we shall explain in Section 6 how to extend the study to Hausdorff measures and large intersection classes associated with general gauge functions
Before addressing all these topics, we remind the reader of elementary results on Diophantine approximation, and basic notions about Hausdorff measures and dimension; this is the purpose of Sections 2 and 3
Summary
The aim of these notes is to present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type Such sets arise naturally in Diophantine approximation and multifractal analysis. Under natural hypotheses on the system, a general construction called ubiquity will enable us to derive the Hausdorff dimension of the latter set, and to show that the large intersection property holds, see Sections 4 and 5. Before addressing all these topics, we remind the reader of elementary results on Diophantine approximation, and basic notions about Hausdorff measures and dimension; this is the purpose of Sections 2 and 3
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