The singular part of Borel transform of a QCD amplitude near the infrared renormalon can be expanded in terms of higher order Wilson coefficients of the operators associated with the renormalon. In this paper we observe that this expansion gives nontrivial constraints on the Borel amplitude that can be used to improve the accuracy of the ordinary perturbative expansion of the Borel amplitude. I...

Suppose that X is a standard Borel space, E is a countable Borel equivalence relation on X, and μ is an E-invariant Borel probability measure on X. We consider the circumstances under which for every countable non-abelian free group Γ, there is a Borel sequence (·r)r∈R of free actions of Γ on X, generating subequivalence relations Er of E with respect to which μ is ergodic, with the further pro...

We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinator...

We establish the generic inexistence of stationary Borel probability measures for aperiodic Borel actions of countable groups on Polish spaces. Using this, we show that every aperiodic continuous action of a countable group on a compact Polish space has an invariant Borel set on which it has no σ-compact realization.

There are continuum many ≤B-incomparable equivalence relations induced by a free, Borel action of a countable non-abelian free group – and hence, there are 20 many treeable countable Borel equivalence relations which are incomparable in the ordering of Borel reducibility

We introduce a new method, involving infinite games and Borel determinacy, which we use to answer several well-known questions in Borel combinatorics.