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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


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 = 0. I will highlight how the analysis seems to critically depend on x₀, 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 https://arxiv.org/abs/2309.08114 in collaboration with Valeria Banica (Sorbonne Universite), Andrea Nahmod (University of Massachusetts) and Luis Vega (BCAM, UPV/EHU).