The Thomae Function: Fractal Insights

TL;DR

The paper analyzes the Thomae function $f_\theta$, a canonical example of a function continuous on irrationals and discontinuous at rationals, by establishing its Hölder regularity in terms of the irrationality exponent $\tau(x)$. Using continued fraction convergents and Diophantine approximation, it proves the pointwise exponent $H_{f_\theta}(x)=0$ for rational $x$ and $H_{f_\theta}(x)=\theta/\tau(x)$ for irrational $x$, and derives a full Hölder spectrum $h\mapsto 2h/\theta$ on $[0,\theta/2]$. The work also discusses differentiability thresholds, ties to Brjuno-type functions, and Boyd-function generalizations, highlighting a multifractal, arithmetic-driven regularity landscape. Overall, it links number-theoretic approximation properties to analytic regularity, offering a precise, fractal perspective on a classical irregular function and guiding generalizations to broader function families.

Abstract

This article examines the Thomae function, a paradigmatic example of a function that is continuous on the irrationals and discontinuous elsewhere. Defined for a parameter $θ>0$, it exhibits a rich self-similar structure and intriguing regularity properties. After revisiting its fundamental characteristics, we analyze its Hölder continuity, emphasizing the interplay between its discrete spikes and its behavior on dense subsets of the real line. This study provides a refined perspective on the irregularity of the Thomae function, using classical analytical tools to elucidate its fractal nature.

Figures (2)

• Figure 1: Representation of the function $f_\theta$ on $(0,1)$ for $\theta=1/2$, $1$ and $2$.
• Figure 2: Representation of the function $f_\phi$ on $(0,1)$ with $\phi(x)=t\ln(1/x)$.

Theorems & Definitions (15)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 4.1