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On the Wasserstein distance between classical sequences and the Lebesgue measure
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 $ 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
$\displaystyle \left \vert \int _{\mathbb{T}^d} f(x) dx - \frac {1}{N} \sum _{k=... ...bla f\Vert^{(d-1)/d}_{L^{\infty }}\Vert \nabla f\Vert^{1/d}_{L^{2}} }{N^{1/d}}.$

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, $ \Vert \nabla f\Vert _{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.
更新日期:2020-11-21
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