Transactions of the American Mathematical Society ( IF 1.363 ) Pub Date : 2020-10-05 , DOI: 10.1090/tran/8212
Louis Brown; Stefan Steinerberger

Abstract:We discuss the classical problem of measuring the regularity of distribution of sets of points in . A recent line of investigation is to study the cost ( mass distance) necessary to move Dirac measures placed on these points to the uniform distribution. We show that Kronecker sequences satisfy optimal transport distance in dimensions. This shows that for differentiable and badly approximable vectors , we have

We note that the result is uniform in (it holds for a sequence instead of a set). Simultaneously, it refines the classical integration error for Lipschitz functions, . 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.

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