A Radon measure \(\mu \) is n-rectifiable if it is absolutely continuous with respect to \({\mathcal {H}}^n\) and \(\mu \)-almost all of \({{\,\mathrm{supp}\,}}\mu \) can be covered by Lipschitz images of \({\mathbb {R}}^n\). In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called \(\alpha \) and \(\beta _2\) numbers. The second one involves \(\alpha _2\) numbers—coefficients quantifying flatness via Wasserstein distance \(W_2\). Both conditions are necessary for rectifiability, too—the first one was shown to be necessary by Tolsa, while the necessity of the \(\alpha _2\) condition is established in our recent paper. Thus, we get two new characterizations of rectifiability.

甲Radon测度\（\亩\）是Ñ -rectifiable如果它是相对于绝对连续\（{\ mathcal {H}} ^ N \）和\（\亩\） -almost所有的\（{{\ ，\ mathrm {supp} \，}} \ mu \）可以被\（{\ mathbb {R}} ^ n \）的Lipschitz图像覆盖。本文中，就平坦度量化系数的平方函数而言，我们给出了两个可矫正的充分条件。第一个条件涉及所谓的\（\ alpha \）和\（\ beta _2 \）数字。第二个涉及\（\ alpha _2 \）个数字-通过Wasserstein距离\（W_2 \）量化平面度的系数。这两个条件也是可纠正性所必需的-Tolsa已证明第一个条件是必需的，而我们最近的论文中确定了\（\ alpha _2 \）条件的必要性。因此，我们得到了可纠正性的两个新特征。