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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 2016

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On Small Subsets in Euclidean Spaces

Mycielski J., Tomkowicz G.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 2/3

109--118

We study a property of smallness of sets which is stronger than the possibility of packing the set into arbitrarily small balls (i.e., being Tarski null) but weaker than paradoxical decomposability (i.e., being a disjoint union of two sets equivalent by finite decomposition to the whole). We show, using the Axiom of Choice for uncountable families, that there are Tarski null sets which are not small sets. Using only the Principle of Dependent Choices, we show that bounded subsets of Rn that are included in countable unions of proper analytic subsets of Rn are small, and several related results.

On Intersections of Generic Perturbations of Definable Sets

Mycielski J., Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 2/3

95--103

Consider an o-minimal expansion R of a real closed field R and two definable sets E and M. We introduce concepts of a locally transitive (abbreviated to l.t.) and a strongly locally transitive (abbreviated to s.l.t.) action of E on M. In the former case, M is supposed to be of pure dimension m; in the latter, both M and E are supposed to be of pure dimension. We treat the elements of E as perturbations of the set M. We prove that if E acts l.t. on M, and A and B are two non-empty definable subsets of M of dimension dim A≤ dim B < dim M, then dim(σ(A) ∩ B) < dim A for a generic σ in E; here dim ∅ = −1. And if E acts s.l.t. on M and A and B are two definable subsets of M, then dim(σ(A) ∩ B) ≤ max{dim A + dim B − m, −1} for a generic σ in E. We give an example of a l.t. action E on M for which the latter conclusion of the intersection theorem fails. We also prove a theorem on the intersections of generic perturbations in terms of the exceptional set T ⊂ M of points at which E is not l.t. Finally, we provide some natural conditions which imply that T is a nowhere dense subset of M.