Related papers: Measure reducibility of countable Borel equivalenc…

Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are…

We consider an old question of Slaman and Steel: whether Turing equivalence is an increasing union of Borel equivalence relations none of which contain a uniformly computable infinite sequence. We show this question is deeply connected to…

We present a general method of constructing an uncountable family of regular Borel measures on certain path spaces of Lipschitz functions having fixed Lipschitz constants. We use this method to give a definition of Lebesgue measure and…

We prove a analog of Kurosh theorem for countable Borel equivalence relations giving the structure of sub-relations in free products.

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of…

We study a uniform, quantitative form of the amenability-hyperfiniteness paradigm for bounded-degree Borel graphs generating countable Borel equivalence relations. We introduce \emph{uniform Borel amenability} and prove that it is…

In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let $\mu$ be a finite Borel measure on $\mathbb{R}^n$ and let $0 < m < n$ be an integer. We show that, under the sole assumption…

We first show that in the function realizability topos every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every $T_0$-space is separable and every…

We consider the reverse math strength of the statement $\mathsf{C\text-DM}$:"Every completely determined Borel set is measurable." Over $\mathsf{WWKL}_0$, we obtain the following results analogous to the previously studied category case.…

We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $r$ expansions, and their various…

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to…

For sets $A, B\subset \mathbb N$, their sumset is $A + B := \{a+b: a\in A, b\in B\}$. If we cannot write a set $C$ as $C = A+B$ with $|A|, |B|\geq 2$, then we say that $C$ is $\textit{irreducible}$. The question of whether a given set $C$…

We show that the category of countable Borel equivalence relations (CBERs) is dually equivalent to the category of countable $\mathcal{L}_{\omega_1\omega}$ theories which admit a one-sorted interpretation of a particular theory we call…

We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect…

The present article is part of a research program the aim of which is to find all indecomposable solvable extensions of a given class of nilpotent Lie algebras. Specifically in this article we consider a nilpotent Lie algebra n that is…

For $N \geq 2$, we study the structure of definable abelian group extensions of the additive group $(\mathbb{R}^N,+)$ by countable abelian (Borel) groups $G$. Given an extension $H$ of $(\mathbb{R}^N,+)$ by $G$, we measure the definability…

We construct ergodic discrete probability measure preserving equivalence relations $\cR$ that has no proper ergodic normal subequivalence relations and no proper ergodic finite-index subequivalence relations. We show that every treeable…

We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the 9 finite graphs from the classical result of Beineke together with a 10th infinite graph associated to the equivalence relation $\mathbb{E}_0$ on the…

We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti…

A framework to handle tree decompositions of the components of a Borel graph in a Borel fashion is introduced, along the lines of Tserunyan's Stallings Theorem for equivalence relations arXiv:1805.09506. This setting leads to a notion of…