We show that the isotropic discrepancy of a lattice point set can be bounded from below and from above in terms of the spectral test of the corresponding integration lattice. From this result we deduce that the isotropic discrepancy of any $N$-element lattice point set in [0,1)${}^d$ is at least of order $N^{-1∕d}$. This order of magnitude is best possible for lattice point sets in dimension $d$.

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关于晶格点集的各向同性差异和谱测试的注释

我们表明，根据相应积分晶格的光谱测试，晶格点集的各向同性差异可以从下方和上方进行界定。根据这个结果，我们可以得出$ñ$-[0,1）中设置的元素晶格点${}^d$ 至少是有秩序的 ${ñ}^{-\mathrm{1个}∕d}$。对于尺寸上的晶格点集，此数量级最可能$d$。