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Multifractality in the evolution of vortex filaments

Multifractality in the evolution of vortex filaments

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

$$

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 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).

## Department of Mathematics and Statistics