Related papers: Borel sets without perfectly many overlapping tran…
The following is true in the Solovay model. 1. If $\leq$ is a Borel partial quasi-order on a Borel set $D$ of the reals, $X$ is a ROD subset of $D$, and $\leq$ restricted to $X$ is linear, then $X$ is countably cofinal in the sense of…
We prove that any incomplete system of complex exponentials $\{e^{i\lambda_n t}\}$ in $L^2(-\pi,\pi)$ is a subset of some complete and minimal system of exponentials. In addition, we prove analogous statement for systems of reproducing…
We resolve the topological version of the Erd\H{o}s Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a…
The inconsistencies involved in the foundation of set theory were invariably caused by infinity and self-reference; and only with the opportune axiomatic restrictions could them be obviated. Throughout history, both concepts have proved to…
Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We…
Generalizing a result of S.V. Konyagin and D.R. Heath--Brown, we prove, in particular, that for any multiplicative subgroup R of Z/pZ and any nonzero elements mu_1,...,mu_k the following holds |R \cap (R+mu_1) \cap ... \cap (R+mu_k)| \ll_k…
In this work we investigate the question of constructions of the possible Fourier bases $E(\Lambda)=\{e^{2\pi i \lambda x}:\lambda\in\Lambda\}$ for the Hilbert space $L^2(\mu_4)$, where $\mu_4$ is the standard middle-fourth Cantor measure…
In this paper, we shall prove that any subset of $\overline{\mathbb Q}\cap B(0,1)$, which is closed under complex conjugation and which contains the element $0$, is the exceptional set of uncountably many transcendental functions, analytic…
In this paper we discuss several variations and generalizations of the Cantor set and study some of their properties. Also for each of those generalizations a Cantor-like function can be constructed from the set. We will discuss briefly the…
Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, pi-character, hereditary Lindelof number, hereditary density. It is shown to be consistent that there exists a sequence…
We construct a function that lies in $L^p(\mathbb{R}^d)$ for every $p \in (1,\infty]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kova\v{c}'s maximal restriction principle to show…
We introduce a new invariant of Borel reducibility, namely the notion of thickness; this associates to every sentence $\Phi$ of $\mathcal{L}_{\omega_1 \omega}$ and to every cardinal $\lambda$, the thickness $\tau(\Phi, \lambda)$ of $\Phi$…
Our main result is a construction of four families C_1,C_2,B_1,B_2 which are equipollent with the power set of the real line R and satisfy the following properties. (i) The members of the families are proper subfields of R whose algebraic…
A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a…
We consider reducibility of equivalence relations (ERs, for brevity), in a nonstandard domain, in terms of the Borel reducibility and the countably determined (CD, for brevity) reducibility. This reveals phenomena partially analogous to…
We show that if \kappa\ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size \kappa\ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2^\kappa\…
Let $C(\lambda )\subset \lbrack 0,1]$ denote the central Cantor set generated by a sequence $ \lambda = \left( \lambda_{n} \right) \in \left( 0,\frac{1}{2} \right) ^{\mathbb{N}}$. By the known trichotomy, the difference set $ C(\lambda…
Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized…
The main question here is the possible generalization of the following theorem on ``simple'' equivalence relation on 2^omega to higher cardinals. Theorem: (1) Assume that: (a) E is a Borel 2-place relation on 2^omega, (b) E is an…
We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the R\'enyi system with one real base. We focus on the so-called…