Describability via ubiquity and eutaxy in Diophantine approximation

TL;DR

This paper develops a unified framework for analyzing the size and intersection properties of sets in Diophantine approximation, integrating classical techniques and applying to various approximation and multifractal contexts.

Contribution

It introduces a comprehensive approach that combines ubiquity and eutaxy concepts, extending classical methods to a broader class of sets and applications in Diophantine approximation.

Findings

Unified description of Hausdorff measures and large intersection classes

Application to classical and inhomogeneous Diophantine approximation

Framework encompasses multifractal analysis of Lévy processes

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 L{\'e}vy processes.