Borel sets with large squares

Borel Sets with Large Squares

For a cardinal μ we give a sufficient condition ⊕μ (involving ranks measuring existence of independent sets) for: ⊗μ: if a Borel set B ⊆ R × R contains a μ-square (i.e. a set of the form A × A, |A| = μ) then it contains a 20 -square and even a perfect square. And also for ⊗′μ: if ψ ∈ Lω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute...

Borel Classes of Uniformizations of Sets with Large Sections

We give several refinements of known theorems on Borel uniformizations of sets with ”large sections”. In particular, we show that a set B ⊂ [0, 1] × [0, 1] which belongs to Σ0α, α ≥ 2, and which has all ”vertical” sections of positive Lebesgue measure, has a Π0α uniformization which is the graph of a Σ0α-measurable mapping. We get a similar result for sets with nonmeager sections. As a corollar...

Lattice Points in Large Borel Sets and Successive Minima

Let B be a Borel set in E with volume V (B) = ∞. It is shown that almost all lattices L in E contain infinitely many pairwise disjoint d-tuples, that is sets of d linearly independent points in B. A consequence of this result is the following: let S be a star body in E with V (S) = ∞. Then for almost all lattices L in E the successive minima λ1(S,L), . . . , λd(S,L) of S with respect to L are 0...

A Course on Borel Sets

Generating Borel Sets by Balls

It is proved that an arbitrary infinite-dimensional Banach space with basis admits an equivalent norm such that any Borel set can be obtained from balls by taking complements and countable disjoint unions. For reflexive spaces, the new norm can be chosen arbitrarily close to the initial norm.