Multifractal phenomena and packing dimension

TL;DR

This paper develops a general framework for understanding multifractal phenomena in functions, deriving new results and extending them to Fourier and Dirichlet series through careful constructions.

Contribution

It introduces an abstract approach to multifractal analysis that simplifies deriving results and extends applicability to Fourier and Dirichlet series.

Findings

Multiple types of multifractal functions can be characterized using the new framework.

The approach yields new results for functions outside traditional multifractal analysis, such as Fourier and Dirichlet series.

Explicit constructions demonstrate the extension of the theory to these cases.

Abstract

We undertake a general study of multifractal phenomena for functions. We show that the existence of several kinds of multifractal functions can be easily deduced from an abstract statement, leading to new results. This general approach does not work for Fourier or Dirichlet series. Using careful constructions, we extend our results to these cases.