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 on these points to the uniform distribution. We show that Kronecker sequences satisfy optimal transport distance in $d \geq 2$ dimensions. This shows that for differentiable $f: \mathbb {T}^d \rightarrow \mathbb {R}$ and badly approximable vectors $\alpha \in \mathbb {R}^d$, we have \begin{equation*} \left | \int _{\mathbb {T}^d} f(x) dx - \frac {1}{N} \sum _{k=1}^{N} f(k \alpha ) \right | \leq c_{\alpha } \frac { \| \nabla f\|^{(d-1)/d}_{L^{\infty }}\| \nabla f\|^{1/d}_{L^{2}} }{N^{1/d}}. \end{equation*} We note that the result is uniform in $N$ (it holds for a sequence instead of a set). Simultaneously, it refines the classical integration error for Lipschitz functions, $\| \nabla f\|_{L^{\infty }} N^{-1/d}$. 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 conveniently ‘recycled’. We present several open problems.