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 . 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 . 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.
Accepted:
Published online:
Daniel Eceizabarrena 1
@article{CRMATH_2019__357_11-12_846_0, author = {Daniel Eceizabarrena}, title = {Some geometric properties of {Riemann's} non-differentiable function}, journal = {Comptes Rendus. Math\'ematique}, pages = {846--850}, publisher = {Elsevier}, volume = {357}, number = {11-12}, year = {2019}, doi = {10.1016/j.crma.2019.10.007}, language = {en}, }
Daniel Eceizabarrena. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.10.007/
[1] Differentiability and dimension of some fractal Fourier series, Adv. Math., Volume 142 (1999) no. 2, pp. 335-354 | DOI
[2] Some Fourier series with gaps, J. Anal. Math., Volume 101 (2007), pp. 179-197 | DOI
[3] Multifractal behaviour of polynomial Fourier series, Adv. Math., Volume 250 (2014), pp. 1-34 | DOI
[4] Vortex filament equation for a regular polygon, Nonlinearity, Volume 27 (2014), pp. 3031-3057 | DOI
[5] Selfsimilarity of Riemann's nondifferentiable function, Nieuw Arch. Wiskd. (4), Volume 9 (1991) no. 3, pp. 303-337
[6] Asymptotic behaviour and Hausdorff dimension of Riemann's non-differentiable function (preprint) | arXiv
[7] Geometric differentiability of Riemann's non-differentiable function (preprint) | arXiv
[8] The differentiability of the Riemann function at certain rational multiples of π, Amer. J. Math., Volume 92 (1970), pp. 33-55 | DOI
[9] More on the differentiability of the Riemann function, Amer. J. Math., Volume 93 (1971), pp. 33-41 | DOI
[10] Weierstrass' non-differentiable function, Trans. Amer. Math. Soc., Volume 17 (1915) no. 3, pp. 301-325 | DOI
[11] Some problems of Diophantine approximations (II), Acta Math., Volume 37 (1914), pp. 193-239 | DOI
[12] Pointwise analysis of Riemann's “nondifferentiable” function, Invent. Math., Volume 105 (1991) no. 1, pp. 157-175 | DOI
[13] The spectrum of singularities of Riemann's function, Rev. Mat. Iberoam., Volume 12 (1996) no. 2, pp. 441-460 | DOI
[14] On the regularity of fractional integral of modular forms, Trans. Amer. Math. Soc., Volume 372 (2019) no. 2, pp. 829-857 | DOI
[15] Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen, Math. Werke II, Königl. Akad. Wiss., 1895, pp. 71-74
Cited by Sources:
Comments - Policy