Mathematical analysis
Some geometric properties of Riemann's non-differentiable function
Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 846-850.

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 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.

La célèbre fonction non différentiable de Riemann est une fonction continue, mais presque nulle part dérivable. Des simulations numériques montrent qu'une de ses versions complexes représente une trajectoire temporelle dans le cadre de l'équation du flot binormal, aussi connue sous le nom de Vortex Filament Equation. Par conséquent, on analyse certaines propriétés géométriques de son image dans C. Dans cette note, on affirme que la dimension de Hausdorff de l'image n'est jamais plus grande que 4/3 et qu'elle n'a pas de tangentes.

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Accepted:
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DOI: 10.1016/j.crma.2019.10.007
Eceizabarrena, Daniel 1

1 BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain
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Eceizabarrena, Daniel. Some geometric properties of Riemann's non-differentiable function. Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 846-850. doi : 10.1016/j.crma.2019.10.007. http://www.numdam.org/articles/10.1016/j.crma.2019.10.007/

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