Vortex filaments that evolve according the binormal floware expected to exhibit turbulent properties.Aiming to quantify this, I will discuss the multifractal properties of the family of functions

$$R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2},\qquad x_0 \in [0,1],$$

that approximate the trajectories of regular polygonal vortex filaments. These functions are a generalization of the classical Riemann's non-differentiable function, which we recover when $x_0 = 0$. I will highlight how the analysis seems to critically depend on $x_0$, and I will discuss the important role played by Gauss sums, a restricted version of Diophantine approximation, the Duffin-Schaeffer theorem, and the mass transference principle. This talk is based on the article \url{https://arxiv.org/abs/2309.08114} in collaboration with Valeria Banica (Sorbonne Universit\'e), Andrea Nahmod (University of Massachusetts) and Luis Vega (BCAM, UPV/EHU).