Related papers: On a topological Erd\H{o}s similarity problem

In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which $\aleph_{\omega+1}$ carries a uniform ultrafilter that is $\theta$-indecomposable for every uncountable cardinal $\theta<\aleph_\omega$. In this…

In this paper we shall show that there exists a polynomial unimodal map f: [0,1] -> [0,1] which is 1) non-renormalizable(therefore for each x from a residual set, $\omega(x)$ is equal to an interval), 2) for which $\omega(c)$ is a Cantor…

We investigate variants of Marstrand's projection theorem that hold for sets of directions and classes of sets in $\mathbb{R}^2$. We say that a set of directions $D \subseteq\mathcal{S}^1$ is $\textit{universal}$ for a class of sets if, for…

The embedding theorem of Roelcke and Dierolf for the completions of four standard uniform structures on topological groups and their quotients holds more generally for spaces of uniform measures. The natural mappings between the four spaces…

We show that in $\mathbb{R}^d$ there are purely unrectifiable sets of Hausdorff (and even box counting) dimension $d-1$ which are not tube null, settling a question of Carbery, Soria and Vargas, and improving a number of results by the same…

Hadwiger's theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent…

The basic theme of this paper is the fact that if $A$ is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd\H os-Szemer\'edi [E-S]. (see also…

We construct a class of homogeneous Cantor-Moran measures with all contraction ratios being reciprocal of integers, and prove that they are pointwise absolutely normal. Our approach relies on methods developed by Davenport, Erd{\H{o}}s, and…

Let $D$ be the ring of $S$-integers in a global field and $\hat{D}$ its profinite completion. We discuss the relation between density in $D$ and the Haar measure of $\hat{D}$: in particular, we ask when the density of a subset $X$ of $D$ is…

In 1994, John Cobb asked: given $N>m>k>0$, does there exist a Cantor set in $\mathbb R^N$ such that each of its projections into $m$-planes is exactly $k$-dimensional? Such sets were described for $(N,m,k)=(2,1,1)$ by L.Antoine (1924) and…

We show that the existence of a universal structure implies the existence of a generic structure for any approximable class $\mathcal{C}$ of countable structures. We also show that the converse is not true. As a consequence, we provide…

We consider generalized Wigner ensembles and general beta-ensembles with analytic potentials for any beta larger than 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk…

Since their introduction by Erd\H{o}s in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding…

We provide two simple universality criteria. Our first criterion states that $\mathcal{S}\subset G_d:=U(d)$ is universal if and only if $\mathcal{S}$ forms a $\delta$-approximate $t(d)$-design, where $t(2)=6$ and $t(d)=4$ for $d\geq3$. Our…

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is of type 1 if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative…

Universal cover in $\mathbb{E}^{n}$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover…

An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We…

We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of…

Discussions surrounding the nature of the infinite in mathematics have been underway for two millennia. Mathematicians, philosophers, and theologians have all taken part. The basic question has been whether the infinite exists only in…

Furstenberg has associated to every topological group $G$ a universal boundary $\partial(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\partial(G,H)$. In the case…