A Borel equivalence relation on a Polish space is countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence rel...

We establish quantitative results for the statistical behaviour of infinite systems. We consider two kinds infinite system: style='text-indent:20px;'>ⅰ) a conservative dynamical system \begin{document}$ (f,X,\mu) $\end{document} preserving id="M3">\begin{document}$ \sigma $\end{document}...

For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We study in this paper the global structure of the classes of K-structurable equivalence relations for various K. We show that K-structurability interacts well with several kinds of Borel homomorphism...

We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree a...

For any countable Borel equivalence relation E on a standard Borel space X, there is a Borel function θ from X to the 2-generated groups such that xEy ⇔ θ(x) ∼= θ(y) .

The perturbative expansions in weak coupling constant in quantum field theories are in general divergent, with the coefficients growing in factorial. One way to make a rigorous sense out of the divergent series is to resum the series using Borel transform. When the large order behavior of the expansion is sign-alternating Borel resummation may yield the full amplitude. But in QCD the series are...

We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures. Introduction. The Borel reducibility theory of Pol...

It is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be Borel summable to the solutions themselves appear also to be the unique solutions to the 1D Schrödinger equation having this property. Namely, it is shown in this paper that for the polynomial potentials the Borel function defined by the fundamental solutions can be conside...