Small sets containing any pattern

MOLTER, URSULA; YAVICOLI, ALEXIA

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY

CAMBRIDGE UNIV PRESS

Lugar: Cambridge; Año: 2020 vol. 168 p. 57 - 73

0305-0041

Given any dimension function $h$, we construct a perfect set $E subseteq R$ 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 $R^N$, of $h$-Hausdorff measure zero, such that for any finite set ${ f_1,ldots,f_n}subseteq mathcal{F}$, $E$ satisfies that $igcap_{i=1}^n f^{-1}_i(E)eqemptyset$.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.