Given any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set E in ${\mathbb{R}}^N$, of h-Hausdorff measure zero, such that for any finite set {f1,. . .,fn} ⊆ $\mathcal{F}$, E satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$.We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points.

中文翻译：

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包含任何模式的小集合

给定任何维度函数H, 我们构造一个完美的集合乙⊆${\mathbb{R}}$零H-Hausdorff 测度，包含任何有限多项式模式。这是作为更一般构造的特例实现的，其中我们有一系列函数$\数学{F}$满足某些条件，我们构造一个完美的集合乙在${\mathbb{R}}^N$， 的H-Hausdorff 测度为零，使得对于任何有限集 {F1,. . .,Fn} ⊆$\数学{F}$,乙满足$\bigcap_{i=1}^nf^{-1}_i(E)\neq\emptyset$.对于函数的图像，我们也得到了类似的结果。此外，我们证明了可数（不一定是有限）交集的一些相关结果，获得，而不是完美集合，$\mathcal{F}_{\sigma}$设置没有孤立点。