  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.
