Related papers: Borel sets without perfectly many overlapping tran…

There exists a family $\{B_{\alpha}\}_{\alpha<\omega_1}$ of sets of countable ordinals such that o $\max B_{\alpha}=\alpha$, o if $\alpha\in B_{\beta}$ then $B_{\alpha}\subseteq B_{\beta}$, o if $\lambda\leq \alpha$ and $\lambda$ is a limit…

For which infinite cardinals $\kappa$ is there a partition of the real line $\mathbb R$ into precisely $\kappa$ Borel sets? Hausdorff famously proved that there is a partition of $\mathbb R$ into $\aleph_1$ Borel sets. But other than this,…

Let $\mathcal X$ be an infinite locally compact separable metric space with metric $\rho$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $\beta = \sup \big\{ \rho(x, y): \{x, y \} \subset \mathcal…

We show that every Banach space $X$ containing an isomorphic copy of $c_0$ has an infinite equilateral set and also that if $X$ has a bounded biorthogonal system of size $\alpha$ then it can be renormed so as to admit an equilateral set of…

Given a compact interval $I \subseteq \mathbb{R}$, and a function $f$ that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates $\{ f(\cdot - \lambda) : \lambda \in \Lambda \}$ are complete in $C(I)$ if…

We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a…

We show that a circle and square of the same area in $\mathbb{R}^2$ are equidecomposable by translations using $\mathbf{\Delta}^0_2$ pieces. That is, pieces which are simultaneously $F_\sigma$ and $G_\delta$ sets. This improves a result of…

Let xi be a non-null countable ordinal. We study the Borel subsets of the plane that can be made $\bormxi$ by refining the Polish topology on the real line. These sets are called potentially $\bormxi$. We give a Hurewicz-like test to…

Let $A$ and $B$ be Borel subsets of the Euclidean $n$-space with $\dim A + \dim B > n$. This is a survey on the question: what can we say about the Hausdorff dimension of the intersections $A\cap (g(B)+z)$ for generic orthogonal…

As a variant of the equal entropy cover problem, we ask whether all multidimensional sofic shifts with countably many configurations have SFT covers with countably many configurations. We answer this question in the negative by presenting…

In the present article we describe how one can define Hausdorff measure allowing empty elements in coverings, and using infinite countable coverings only. In addition, we discuss how the use of different nonequivalent interpretations of the…

We show that if $E$ is a countable Borel equivalence relation on $\mathbb{R}^n$, then there is a closed subset $A \subset [0,1]^n$ of Hausdorff dimension $n$ so that $E \restriction A$ is smooth. More generally, if $\leq_Q$ is a locally…

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if B is a G-delta-sigma set, then either B is countable or B contains a perfect subset. Second, we…

We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.

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…

Let X be an uncountable Polish space. Lubica Hola showed recently that there are 2^continuum many quasi-continuous real valued functions defined on the uncountable Polish space that are not Borel measurable. Inspired by Hola's result, we…

We study the conformal bootstrap for systems of correlators involving non-identical operators. The constraints of crossing symmetry and unitarity for such mixed correlators can be phrased in the language of semidefinite programming. We…

Assuming that $0^\dagger$ does not exist, we prove that if there is a partition of $\mathbb R$ into $\aleph_\omega$ Borel sets, then there is also a partition of $\mathbb R$ into $\aleph_{\omega+1}$ Borel sets.

We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces.

We investigate variants of the Erd\H{o}s similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known…