Wasserstein distance, Fourier series and applications

Abstract

We study the Wasserstein metric $W_p$, a notion of distance between two probability distributions, from the perspective of Fourier Analysis and discuss applications. In particular, we bound the Earth Mover Distance $W_1$ between the distribution of quadratic residues in a finite field ${\mathbb {F}}_p$ and uniform distribution by $\lesssim p^{-1/2}$ (the Polya–Vinogradov inequality implies $\lesssim p^{-1/2} \log {p}$). We also show that for continuous $f:{\mathbb {T}} \rightarrow {\mathbb {R}}_{}$ with mean value 0

$$\begin{aligned} (\text{ number } \text{ of } \text{ roots } \text{ of }~f) \cdot \left( \sum _{k=1}^{\infty }{ \frac{ |{\widehat{f}}(k)|^2}{k^2}}\right) ^{\frac{1}{2}} > rsim \frac{\Vert f\Vert ^{2}_{L^1({\mathbb {T}})}}{\Vert f\Vert _{L^{\infty }({\mathbb {T}})}}. \end{aligned}$$

Moreover, we show that for a Laplacian eigenfunction $-\Delta _g \phi _{\lambda } = \lambda \phi _{\lambda }$ on a compact Riemannian manifold $W_p\left( \max \left\{ \phi _{\lambda }, 0\right\} dx, \max \left\{ -\phi _{\lambda }, 0\right\} dx\right) \lesssim _p \sqrt{\log {\lambda }/\lambda } \Vert \phi _{\lambda }\Vert _{L^1}^{1/p}$, which is at most a factor $\sqrt{\log {\lambda }}$ away from sharp. Several other problems are discussed.