Hölder Regularity and Fractal Aspects of the Thomae Function

Voici le titre : Hölder Regularity and Fractal Aspects of the Thomae Function.
Voici le résumé : The Thomae function has long served as a striking example in real analysis, showcasing the interplay between continuity and discontinuity. Introduced by Thomae in 1875 as a refinement of the Dirichlet function, it is defined as follows. Unless explicitly stated otherwise, any rational number $x$ expressed as $x=p/q$ ($p\in \Z$, $q\in \N$) with $p$ and $q$ coprime. The Thomae function is then given by
\[
T_\theta (x) =
\left\{\begin{tabular}{ll}
$1$ & if $x=0,$ \\
$q^{-\theta}$ & if $x$ is rational with $x=p/q,$ \\
$0$ & if $x$ is irrational,
\end{tabular}\right.
\]
with $\theta=1$. The limiting case $\theta=0$ corresponds to the Dirichlet function.
This talk focuses on the H\"older regularity of the Thomae function, a key aspect of its behaviour. First, we review its fundamental properties, offering a detailed account of its defining characteristics and self-similar nature. Then, we analyze the function's H\"older regularity, uncovering insights into its fractal-like properties through contemporary mathematical tools. By bridging its classical foundations with these contemporary perspectives, we aim to highlight both the theoretical elegance and the deeper structural nuances of this remarkable function. We also consider properties of functions that are positive on rationals and 0 on irrationals.