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arXiv · 1610.03804

Small Sets containing any Pattern

Ursula Molter·Alexia Yavicoli
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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 $\mathcal{F}_σ$ set without isolated points.

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Ursula Molter, Alexia Yavicoli. 2018-06-13. Small Sets containing any Pattern. https://doi.org/10.1017/s0305004118000567

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