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

Abstract

Riemann's non-differentiable function, whose analytic regularity has been widely studied, can also be analysed from a geometric perspective. Indeed, it can be generalised to the complex plane to represent the trajectory of a vortex filament in the setting of the binormal flow. We give an upper estimate for the Hausdorff dimension of this fractal-like trajectory when studied as a subset of R 2 . For that, we compute a precise asymptotic behaviour of Riemann's function at rational points.

An analytical study of flatness and intermittency through Riemann's non-differentiable functions

“…Denoting N = 2 k , we see that P k f = f N is a band-pass filter of f . From (15) and the definition of ζ p in ( 14), formally we have f N p p N −ζp , where lower order corrections like logarithms are not accounted for. Also, in many situations like for R s , f N p f ≥N p holds.…”

“…Our use of R s comes from Jaffard proving that Riemann's non-differentiable function R 1 satisfies the multifractal formalism [21]. Riemann's function also appears as the trajectory of polygonal vortex filaments driven by the binormal flow [1,13], and has thus been geometrically studied [9,[14][15][16]. Further generalizations of R s have also been studied [9][10][11].…”

Geometric differentiability of Riemann's non-differentiable function

“…(A) Around φ(t 1/2 ), placed in the centre of the spiral, where precisely the spiral pattern prevents a tangent from forming. As already suggested, the main ingredient in the proof of Theorem 1.1 is the asymptotic behaviour of φ around every rational t p/q proved in [4], which for convenience we state in the forthcoming sections. Hence, tangency around a rational is easy to manage, but a more subtle analysis is required around an irrational.…”

“…One could think of checking the points where the derivative exists, but the results of Gerver [8,9] and (3) show that the corresponding value of the derivative of φ is 0, which is useless to determine a tangent. What is more, one can deduce from the asymptotics in [4] that a spiralling pattern is generated (Figure 2A). Second, the asymptotics in the rest of rationals show that derivative does not exist and that the regularity is C 1/2 , but nevertheless, they suggest the existence of two different geometric tangents at each side (Figure 2B).…”

“…Moreover, both functions being non-injective, it is difficult to measure its effect in the images. Inspired by the works above, especially by [3], in [4] the asymptotic behaviour for φ is given in all points t x = x/2π with x rational. There, a self-similar structure, already visible in Figure 1, is analytically shown.…”