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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.

Intersection of Generic Rotations in Some Classical Spaces

Nowak K., Tomkowicz G.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 2/3

105--107

Consider an o-minimal structure on the real field R and two definable subsets A, B of the Euclidean space Rn, of the unit sphere Sn or of the hyperbolic space Hn, n ≥ 2, which are of dimensions k, l ≤ n−1, respectively. We prove that the dimension of the intersection σ(A) ∩ B is less than min{k, l} for a generic rotation σ of the ambient space; here we set dim ∅ = −1.