Zoltán Vidnyánszky

Haar null sets without G δ hulls

Let G be an abelian Polish group, e.g., a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure µ on G such that µ(B + g) = 0 for every g ∈ G. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent.Answering an old question of Mycielski we show that if G is not locally compact then there exists a Borel Haar null set that is not contained in any

Transfinite inductions producing coanalytic sets

{G_\delta }

Naively Haar null sets in Polish groups

Haar null set. We also show that

The structure of random automorphisms

{G_\delta }

Characterization of order types of pointwise linearly ordered families of Baire class 1 functions

can be replaced by any other class of the Borel hierarchy, which implies that the additivity of the σ-ideal of Haar null sets is ω1.The definition of a generalised Haar null set is obtained by replacing the Borelness of B in the above definition by universal measurability. We give an example of a generalised Haar null set that is not Haar null, more precisely, we construct a coanalytic generalised Haar null set without a Borel Haar null hull. This solves Problem GP from Fremlin’s problem list. Actually, all our results readily generalise to all Polish groups that admit a two-sided invariant metric.

The structure of random automorphisms of countable structures.

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class 1 functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class functions, and generalize most of the results from the Baire class 1 case. As an application, we solve a problem concerning the so called solvability cardinals of systems of dierence equations, arising from the theory of geometric equidecomposability. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in !1.