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.