On the Wasserstein Distance between Classical Sequences and the Lebesgue Measure
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We discuss the classical problem of measuring the regularity of distribution of sets of N points in T^d. A recent line of investigation is to study the cost (= mass × 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 ≥ 3 dimensions. This shows that for differentiable f: T^d →R and badly approximable vectors α∈R^d, we have | ∫_T^d f(x) dx - 1/N∑_k=1^N f(k α) | ≤ c_α∇ f^(d-1)/d_L^∞∇ 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, ∇ f_L^∞ 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.
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