Small Sets containing any Pattern

Ursula Molter, Alexia Yavicoli

Published 2016-10-12Version 1

Given a dimension function $h$, and a family of functions $\mathcal{F}$ that satisfy certain conditions, we construct a closed 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 $\bigcap_{i=1}^n f_i(E)\neq\emptyset$. We obtain an analogous result for the preimages, from which we are able to prove that there exists a closed set in $\mathbb{R}$, of zero $h$-Hausdorff measure, that contains any polynomial pattern. The set $E$ can - in some cases - chosen to be perfect. Additionally we prove some related result for countable intersections, obtaining, instead of a closed set, an $\mathcal{F}_{\sigma}$ set.