Some geometric properties of Riemann's non-differentiable function

Abstract

Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in $\mathbb{C}$. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.

6 pages, 2 figures. v2: References updated and corrected

References in corpus (2)

• Vortex Filament Equation for a Regular Polygon
• Geometric differentiability of Riemann's non-differentiable function

Cited by in corpus (4)

• Geometric differentiability of Riemann's non-differentiable function
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• Intermittency of Riemann's non-differentiable function through the fourth-order flatness
• An analytical study of flatness and intermittency through Riemann's non-differentiable functions