Abstract:

Given any dimension function h, we construct a perfect set E 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 that satisfy certain conditions and we construct a perfect set E in, of h-Hausdorff measure zero, such that for any finite set {f1,.,fn} †, E satisfies that.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 set without isolated points. © 2018 Cambridge Philosophical Society.