Vortex filaments that evolve according the binormal flow are expected to exhibit turbulent properties. Aiming to quantify this, I will discuss the multifractal properties of the family of functions
Rx0(t)=∑n≠0e2πi(n2t+nx0)n2,x0∈[0,1],𝑅𝑥0(𝑡)=∑𝑛≠0𝑒2𝜋𝑖(𝑛2𝑡+𝑛𝑥0)𝑛2,𝑥0∈[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 x0=0𝑥0=0. I will highlight how the analysis seems to critically depend on x0𝑥0, 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 recent article
https://arxiv.org/abs/2309.08114 in collaboration with Valeria Banica (Sorbonne Universite), Andrea Nahmod (University of Massachusetts) and Luis Vega (BCAM, UPV/EHU).