Small Sets containing any Pattern

Abstract

Given any dimension function $h$, we construct a perfect set $E\subseteq \mathbb{R}$ 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 $\{f_1,\dots ,f_n\}\subseteq \mathcal{F}$, $E$ satisfies that ${\bigcap }_{i=1}^n{f}_i^{-1}(E)\ne \varnothing$. 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.

To appear in Mathematical Proceedings of the Cambridge Philosophical Society

References in corpus (3)

• On polynomial configurations in fractal sets
• Squares and their centers
• Large Sets Avoiding Patterns

Cited by in corpus (1)

• On Sets Containing an Affine Copy of Bounded Decreasing Sequences