Related papers: Definable maximal discrete sets in forcing extensi…

Let $\mathcal R$ be a $\Sigma^1_1$ binary relation and call a set $\mathcal R$-discrete iff no two distinct of its elements are $\mathcal R$-related. We show that in the extension of $\mathbf{L}$ by iterated Sacks forcing, there is a…

We study maximal orthogonal families of Borel probability measures on $2^\omega$ (abbreviated m.o. families) and show that there are generic extensions of the constructible universe $L$ in which each of the following holds: (1) There is a…

We prove that if $V=L$ then there is a $\Pi^1_1$ maximal orthogonal (i.e. mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known Theorem of Preiss and Rataj that no analytic set of…

We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, $\mathcal I$-mad families, maximal eventually different families,…

We prove that the generic maximal independent family obtained by iteratively forcing with the Mathias forcing relative to diagonalization filters is densely maximal. Moreover, by choosing the filters with some care one can ensure the family…

We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality $\mathfrak a_{\text{g}}$ of a maximal cofinitary group (MCG) is strictly between $\aleph_1$ and…

We consider Cantor measures on the line, with contraction factor $N^{-1}=p^{-\alpha}$ (where $p$ a positive prime, $\alpha$ a positive integer) and $m$ positive integer digits lying in distinct residue classes modulo $N$. We obtain a…

We present a proof of a result, previously announced by the second author, that there is a closed (even $\Pi^0_1$) set generating an $F_\sigma$ (even $\Sigma^0_2$) maximal cofinitary group (short, mcg) which is isomorphic to a free group.…

We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a $\boldsymbol{\Sigma}^1_2$ m.i.f. is equivalent to the existence of a $\boldsymbol{\Pi}^1_1$ m.i.f., (b) in the Cohen model,…

We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $\omega$. We show that if $\mathcal J$ is an arbitrary $F_\sigma$ ideal, or is any finite or countably iterated Fubini product of $F_\sigma$ ideals,…

A multiplicative subset $S$ of a ring $R$ is called \textit{strongly multiplicative} if $(\bigcap_{i\in\Delta}s_iR)\cap S \neq \emptyset$ for each family $(s_i)_{i\in\Delta}$ of elements in $S$. In this paper, we investigate how these sets…

We show there is a closed (in fact effectively closed, i.e., $\Pi^0_1$) eventually different family (working in ZF or less).

Let $\mathcal{\scriptstyle{O}}_K$ be the ring of integers of an imaginary quadratic number field $K$. In this paper we give a new description of the maximal discrete extension of the group $SL_2(\mathcal{\scriptstyle{O}}_K)$ inside…

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $\phi: M\to\mathbb{R}$ continuous. We prove that there exists a dense subset of $\mathcal{A}$…

We give an explicit description of the restriction of discrete series representations of SU(2,1) to a Borel subgroup and to a maximal exponential solvable subgroup and we interpret our results in the framework of orbit method, Hamiltonian…

In this paper we introduce the notion of enlargement of a positive set in SSD spaces. To a maximally positive set $A$ we associate a family of enlargements $\E(A)$ and characterize the smallest and biggest element in this family with…

We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP$_2$) distributions and log-$L^\#$-concave (LLC) distributions.…

We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…

Let $S = \{p_1,\dots,p_r\}$ be a finite set of distinct primes, let $\Psi_S(X)$ be the number of $S$-smooth integers not exceeding $X$, and let $F_S(X)$ be the maximum size of a subset of $M(S) \cap [1,X]$ containing no set $\{n,p_1…

We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies…