Countable ordinals and the analytical hierarchy, I

The following results are proved, using the axiom of Projective Determinacy: (i) For n ≥ 1, every II(1/2n+1) set of countable ordinals contains a Δ(1/2n+1) ordinal, (ii) For n ≥ 1, the set of reals Δ(1/2n) in an ordinal is equal to the largest countable Σ(1/2n) set and (iii) Every real is Δ(1/n) inside some transitive model of set theory if and only if n ≥ 4

A strong generic ergodicity property of unitary and self-adjoint operators

Consider the conjugacy action of the unitary group of an infinite-dimensional separable Hilbert space on the unitary operators. A strong generic ergodicity property of this action is established, by showing that any conjugacy invariants assigned in a definable way to unitary operators, and taking as values countable structures up to isomorphism, generically trivialize. Similar results are proved for conjugacy of self-adjoint operators and for measure equivalence. The proofs make use of the theory of turbulence for continuous actions of Polish groups, developed by Hjorth. These methods are also used to give a new solution to a problem of Mauldin in measure theory, by showing that any analytic set of pairwise orthogonal measures on the Cantor space is orthogonal to a product measure

Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups

We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.Comment: 73 pages, LaTeX 2e, to appear in Geom. Funct. Ana

A Classification of Baire Class 1 Functions

We study in this paper various ordinal ranks of (bounded) Baire class 1 functions and we show their essential equivalence. This leads to a natural classification of the class of bounded Baire class 1 functions B_1 in a transfinite hierarchy B^ξ_1 ξ < ω_1) of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of Δ^0_2 sets via transfinite differences of closed sets can be carried out. The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class 1 function ƒ are introduced and used in this study

Hausdorff Measures and Sets of Uniqueness for Trigonometric Series

We characterize the closed sets E in the unit circle T which have the property that, for some nondecreasing h: (0, ∞) →(0, ∞) with h(0+) = 0, all the Hausdorff h-measure 0 closed sets F ⊆ E are sets of uniqueness (for trigonometric series). In conjunction with Körner's result on the existence of Helson sets of multiplicity, this implies the existence of closed sets of multiplicity (M-sets) within which Hausdorff h-measure 0 implies uniqueness, for some h. This is contrasted with the case of closed sets of strict multiplicity (M_0-sets), where results of Ivashev-Musatov and Kaufman establish the opposite

The complexity of the classification of Riemann surfaces and complex manifolds

In answer to a question by Becker, Rubel, and Henson, we show that countable subsets of ℂ can be used as complete invariants for Riemann surfaces considered up to conformal equivalence, and that this equivalence relation is itself Borel in a natural Borel structure on the space of all such surfaces. We further proceed to precisely calculate the classification difficulty of this equivalence relation in terms of the modern theory of Borel equivalence relations. On the other hand we show that the analog of Becker, Rubel, and Henson's question has a negative solution in (complex) dimension n ≥ 2

On the theory of ∏_3^1 sets of reals

Assuming that ∀x Є ω^ω (x^# exists), let u_ɑ be the ɑth uniform indiscernible (see [3] or [2] ). A canonical coding system for ordinals < u_ω can be defined by letting W0_ω = {w Є ω^ω: w = (n, x^#), for some n Є ω, x Є ω^ω} and for w = (n, x^#) є W0_ω, │w│ = Ƭ^L_n [x](u_l',... , u_k_n), where T_n is the nth term in a recursive enumeration of all terms in the language of ZF + V = L [x], x a constant, taking always ordinal values. Call a relation P(ξ x), where ~varies over u^ω and x over ω^ω, ∏^1_k if P^*(w, x)⇔ w Є W0_ω Λ P(│w│, x) is ∏^1_k. An ordinal ξ < u_ω is called Δ^1_k if it has a Δ^1_k notation i.e. ∃ w Є W0_ω (w Є Δ^1_k Λ │w│ = ξ)

On Characterizing Spector Classes

We study in this paper characterizations of various interesting classes of relations arising in recursion theory. We first determine which Spector classes on the structure of arithmetic arise from recursion in normal type 2 objects, giving a partial answer to a problem raised by Moschovakis [8], where the notion of Spector class was first essentially introduced. Our result here was independently discovered by S. G. Simpson (see [3]). We conclude our study of Spector classes by examining two simple relations between them and a natural hierarchy to which they give rise

A Basis Result for Σ^0_3 Sets of Reals with an Application to Minimal Covers

It is shown that every ∑^0_3 set of reals which contains reals of arbitrarily high Turing degree in the hyperarithmetic hierarchy contains reals of every Turing degree above the degree of Kleene's O. As an application it is shown that every Turing degree above the Turing degree of Kleene's O is a minimal cover

Subequivalence Relations and Positive-Definite Functions

We study a positive-definite function associated to a measure-preserving equivalence relation on a standard probability space and use it to measure quantitatively the proximity of subequivalence relations. This is combined with a recent co-inducing construction of Epstein to produce new kinds of mixing actions of an arbitrary infinite discrete group and it is also used to show that orbit equivalence of free, measure preserving, mixing actions of non-amenable groups is unclassifiable in a strong sense. Finally, in the case of property (T) groups we discuss connections with invariant percolation on Cayley graphs and the calculation of costs