Small Sets containing any Pattern

TL;DR

The paper constructs perfect sets of zero Hausdorff measure that contain any finite polynomial pattern and extends this to more general function families, with results on intersections and images.

Contribution

It introduces a method to build perfect sets with zero Hausdorff measure containing any finite pattern from a broad class of functions, generalizing previous results.

Findings

Constructed perfect sets with zero $h$-Hausdorff measure containing finite polynomial patterns.

Extended the construction to families of functions satisfying certain conditions.

Proved results on countable intersections leading to $_{ ext{sigma}}$ sets without isolated points.

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 $\{ f_1,\ldots,f_n\}\subseteq \mathcal{F}$, $E$ 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.