On the Wasserstein Distance between Classical Sequences and the Lebesgue Measure

Abstract

We discuss the classical problem of measuring the regularity of distribution of sets of $N$ points in ${\mathbb{T}}^d$. A recent line of investigation is to study the cost ($=$ mass $\times$ distance) necessary to move Dirac measures placed in these points to the uniform distribution. We show that Kronecker sequences satisfy optimal transport distance in $d\ge 3$ dimensions. This shows that for differentiable $f:{\mathbb{T}}^d\to \mathbb{R}$ and badly approximable vectors $\alpha \in {\mathbb{R}}^d$, we have $$|{\int }_{{\mathbb{T}}^d}f(x)dx-\frac{1}{N}\sum _{k=1}^{N}f(k\alpha )|\le c_{\alpha }\frac{\Vert \nabla f{\Vert }_{L^{\infty }}^{(d-1)/d}\Vert \nabla f{\Vert }_{L^2}^{1/d}}{N^{1/d}}.$$ We note that the result is uniformly true for a sequence instead of a set. Simultaneously, it refines the classical integration error for Lipschitz functions, $\Vert \nabla f{\Vert }_{L^{\infty }}{N}^{-1/d}$. We obtain a similar improvement for numerical integration with respect to the regular grid. The main ingredient is an estimate involving Fourier coefficients of a measure; this allows for existing estimates to be conviently `recycled'. We present several open problems.

v2, some minor changes