Related papers: Borel Circle Squaring

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class $\Gamma$, then its $\Gamma$-code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau's theorem to Borel functions: If…

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has…

We expand the results of Roslanowski and Shelah arXive:1806.06283 , arXive:1909.00937 to all perfect Abelian Polish groups $(H,+)$. In particular, we show that if $\alpha<\omega_1$ and $4\leq k<\omega$, then there is a ccc forcing notion…

The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for sigma-finite invariant measures (Corollary 1). For…

We completely determine the $\tau$-tilting finiteness of Borel-Schur algebras. To achieve this, we use two recently introduced techniques in silting theory: sign decomposition as introduced by Aoki, Higashitani, Iyama, Kase and Mizuno…

Given two non-empty subsets $A$ and $B$ of the hyperbolic plane $\mathbb{H}^2$, we define their horocyclic Minkowski sum with parameter $\lambda=1/2$ as the set $[A:B]_{1/2} \subseteq \mathbb{H}^2$ of all midpoints of horocycle curves…

We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$,…

Assume that we are given a coaction \delta of a locally compact group G on a C*-algebra A and a T-valued Borel 2-cocycle \omega on G. Motivated by the approach of Kasprzak to Rieffel's deformation we define a deformation A_\omega of A.…

Let k be an algebraically closed field of characteristic 0, let X=P^1\times A^N and let f be a rational endomorphism of X given by (x,y)--->(g(x), A(x)y), where g is a rational function, while A is an N-by-N matrix with entries in k(x). We…

In the paper we investigate Borel classes of multivalued functions of two variables. In particular we generalize a result of Marczewski and Ryll-Nardzewski concerning of real function whose ones of its sections are right-continuous and…

We establish Borel equivariant analogues of several classical theorems from complex analysis and PDE. The starting point is an equivariant Weierstrass theorem for entire functions: there exists a Borel mapping which assigns to each…

The following will be shown: Let $I$ be a $\sigma$-ideal on a Polish space $X$ with the property that the associated forcing of $I^+$ Borel subsets ordered by $\subseteq$ is a proper forcing. Let E be an analytic or coanalytic equivalence…

For any bounded convex domain \Omega in R^N, we assign a positive finite Borel measure associated with the solution to a su-blinear elliptic equation in \Omega. We prove that this measure is weakly continuous in the sense of measure with…

We give a sketch of proof that any two (Lebesgue) measurable subsets of the unit sphere in $R^n$, for $n\ge 3$, with non-empty interiors and of the same measure are equidecomposable using pieces that are measurable.

We show that every countable Borel equivalence relation structurable by $n$-dimensional contractible simplicial complexes embeds into one which is structurable by such complexes with the further property that each vertex belongs to at most…

\footnotesize B\"{o}r\"{o}czky, Lutwak, Yang and Zhang recently conjectured a certain strengthening of the Brunn-Minkowski inequality for symmetric convex bodies, the so-called log-Brunn-Minkowski inequality. We establish this inequality…

Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets $B$ in $\mathbb{R}$ (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and \sigma-finite measure to $B$. We…

Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This…

If $X$ is an analytic metric space satisfying a very mild doubling condition, then for any finite Borel measure $\mu$ on $X$ there is a set $N\subseteq X$ such that $\mu(N)>0$, an ultrametric space $Z$ and a Lipschitz bijection $\phi:N\to…

Given a set of positive reals, we provide a necessary and sufficient condition for a free Borel flow to admit a cross section with all distances between adjacent points coming from this set.