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On the Hausdorff dimension of Riemann's non-differentiable function

Daniel Eceizabarrena

TL;DR

This paper investigates the geometric roughness of Riemann's non-differentiable function through its vortex-filament interpretation. By deriving precise asymptotics of the associated complex function $\phi$ around rationals and connecting these to Gauss sums and the Talbot effect, the authors establish rigorous bounds on the Hausdorff dimension of the image $\phi(\mathbb{R})$, and extend the results to a multifractal setting. The work integrates modular-theoretic reductions (theta-modular group), continued fractions, and base-case analyses at $0$ and $t_{1,2}$ to reduce the local behavior at general rationals to these canonical cases. The findings provide a first, rigorous upper bound of $4/3$ for the Hausdorff dimension and give a multifractal upper bound $\frac{4\alpha-2}{\alpha}$ for $\alpha\in[\tfrac12,\tfrac34]$, highlighting a structured self-similarity and offering a pathway toward lower bounds via domain multifractality.

Abstract

Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.

On the Hausdorff dimension of Riemann's non-differentiable function

TL;DR

This paper investigates the geometric roughness of Riemann's non-differentiable function through its vortex-filament interpretation. By deriving precise asymptotics of the associated complex function around rationals and connecting these to Gauss sums and the Talbot effect, the authors establish rigorous bounds on the Hausdorff dimension of the image , and extend the results to a multifractal setting. The work integrates modular-theoretic reductions (theta-modular group), continued fractions, and base-case analyses at and to reduce the local behavior at general rationals to these canonical cases. The findings provide a first, rigorous upper bound of for the Hausdorff dimension and give a multifractal upper bound for , highlighting a structured self-similarity and offering a pathway toward lower bounds via domain multifractality.

Abstract

Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.

Paper Structure

This paper contains 27 sections, 16 theorems, 180 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Riemann's non-differentiable function
  3. A physical and geometric version of Riemann's function
  4. Geometric study of Riemann's function
  5. Results
  6. Auxiliary geometric result: the asymptotic behavior of $\phi$ around rationals
  7. Discussion on a lower bound for the Hausdorff dimension
  8. Structure of the document
  9. The Hausdorff dimension
  10. Proof of Theorem \ref{['TheoremHausdorffDimension']}
  11. Proof of Theorem \ref{['TheoremHausdorffDimensionGeneralised']}
  12. Technical results for Section \ref{['Section_HausdorffDimension']}
  13. Proof of the correspondence (\ref{['CorrespondenceJaffard']})
  14. A lemma about continued fractions
  15. The asymptotic behavior: heuristics
  16. ...and 12 more sections

Key Result

Theorem 1

(BanicaVega2020-R) Let $n \in \mathbb N$ and the planar regular polygon of $2n+1$ sides be parametrized by $\boldsymbol{X}_n(s,0)$, which gives a single loop to the polygon when $|s| \leq n$ with its corners located at the integers, and which escapes to infinity by two straight lines when $|s|>n$. T

Figures (4)

  • Figure 1: Numeric simulations of the trajectory of a corner of the $M$-sided regular polygon, for $M=3,4,5$. Image by F. De la Hoz and L. Vega.
  • Figure 2: The set $\phi([0,1/(2\pi)]) \subset \mathbb C$. The resemblance to the numeric trajectories in Figure \ref{['fig:Numeric_Trajectories']} is astonishing.
  • Figure 3: Graphic visualization of the asymptotic behavior of $\phi$ around 0 and $t_{1,2}$. Compare Figure \ref{['fig:Zoom0']} to Figure \ref{['FIG_Curva']} to appreciate the self-similar patterns, which are analytically explained by \ref{['eq:Asymptotic_0_Selfsimilar']} in Proposition \ref{['thm:Asymptotic_At_0']}. In Figure \ref{['fig:Zoom12']}, the spiraling pattern is a consequence of \ref{['eq:Asymptotic_At_12_Short']} in Proposition \ref{['thm:Asymptotic_At_1_2']} and the definition of $Z_1$\ref{['Spiral12']}.
  • Figure 4: Zoom of $\phi(\mathbb R)$ around $\phi(t_{1,8})$. Compare to Figure \ref{['FIG_Curva']} to appreciate the self-similar pattern, which is analytically explained in \ref{['eq:Asymptotic_At_Q013_Selfsimilar']} in Proposition \ref{['thm:Asymptotic_At_Q013']}. Compare it also to the behavior of $\phi$ around 0 in Figure \ref{['fig:Zoom0']}. Except for a rotation by $\pi/4$ radians, they are very similar.

Theorems & Definitions (32)

  • Theorem
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 22 more