On the Wasserstein Distance between Classical Sequences and the Lebesgue Measure

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 $d \geq 3$ dimensions. This shows that for differentiable $f: \mathbb{T}^d \rightarrow \mathbb{R}$ and badly approximable vectors $\alpha \in \mathbb{R}^d$, we have $$\ | \int_{\mathbb{T}^d} f(x) dx - \frac{1}{N} \sum_{k=1}^{N} f(k \alpha) \ | \leq c_{\alpha} \frac{ \| \nabla f\|^{(d-1)/d}_{L^{\infty}}\| \nabla f\|^{1/d}_{L^{2}} }{N^{1/d}}.$$ 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, $\| \nabla f\|_{L^{\infty}} N^{-1/d}$. 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 conviently `recycled'. We present several open problems.