Number Theory Meets Multifractal Analysis

This talk explores the deep interplay between number theory and multifractal
analysis through the study of pointwise regularity of arithmetic functions.
I will begin by presenting the general framework of multifractal analysis
and the notion of p-exponents, a tool that extends classical H¨older regularity
to nonlocally bounded functions. This analytical perspective provides a natural
bridge toward Diophantine approximation, where the irrationality exponent
τ (x) emerges as a key descriptor of local behavior. I will then illustrate these
ideas through the examples of the Brjuno and Thomae functions, whose singularities
and fractal structures reveal precise connections between regularity
exponents and approximation properties of real numbers. Finally, I will discuss
ongoing and prospective research directions, focusing on the Minkowski question
mark function—a singular, strictly increasing function whose pointwise regularity
remains largely mysterious—and how such objects may fit into a unified
multifractal framework driven by number-theoretic dynamics.