A carpet is a metric space that is homeomorphic to the standard Sierpiński carpet in , or equivalently, in S². A carpet is called thin if its Hausdorff dimension is . A metric space is called Q-Loewner if its Q-dimensional Hausdorff measure is Q-Ahlfors regular and if it satisfies a -Poincaré inequality. As we will show, Q-Loewner planar metric spaces are always carpets, and admit quasisymmetric embeddings into the plane.

中文翻译：

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薄 Loewner 地毯及其在 S2 中的准对称嵌入

地毯是一个度量空间，它同胚于 中的标准 Sierpiński 地毯，或等效于S ²中的标准 Sierpiński 地毯。如果地毯的 Hausdorff 维数是 ，则地毯被称为薄地毯。如果一个度量空间的Q维 Hausdorff 测度是 Q-Ahlfors 规则的并且满足-Poincaré 不等式，则该度量空间称为Q-Loewner 。正如我们将要展示的，Q -Loewner 平面度量空间总是地毯，并允许准对称嵌入到平面中。