相关论文: Strong measure zero sets without Cohen reals

If E(z,s) is the nonholomorphic Eisenstein series on the upper half plane, then for all y sufficiently large, E(z,s) has a "Siegel zero." That is E(z,\beta)=0 for a real number \beta just to the left of one. We give a generalization of this…

A sofic measure is the image of a Markov probability measure by a continuous morphism, and can be represented by means of products of matrices $A_n$ that belong to a finite set of nonnegative matrices. To prove that the multifractal…

We show that every null-additive set is meager-additive, where: (1) a set X subseteq 2^omega is null-additive if for every Lebesgue null set A subseteq 2^omega, X+A is null too; (2) we say that X subseteq 2^omega is meager-additive if for…

In this paper, we establish a simple criterion for two $L$-functions $L_1$ and $L_2$ satisfying a functional equation (and some natural assumptions) to have infinitely many distinct zeros. Some related questions have already been answered…

A 1984 problem of S.Z. Ditor asks whether there exists a lattice of cardinality aleph two, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice…

Let P be a set of n points in $\mathbb{R}^d$. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than $dn\over d+1$ points of P. We call a point x a strong centerpoint for a family of…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

We are interested in subgroups of the reals that are small in one and large in another sense. We prove that, in ZFC, there exists a non-meager Lebesgue null subgrooup of R, while it isconsistent there there is no non-null meager subgroup of…

We show that if $cf(2^{\aleph_0})=\aleph_1,$ then any non-trivial $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ is forcing equivalent to $Add(\aleph_1, 1),$ the Cohen forcing for adding a new Cohen subset of $\omega_1.$ We…

We present a negative result regarding the hidden subgroup problem on the powers $G^n$ of a fixed group $G$. Under a condition on the base group $G$, we prove that strong Fourier sampling cannot distinguish some subgroups of $G^n$. Since…

An ultrametric preserving function $f$ is said to be strongly ultrametric preserving if ultrametrics $d$ and $f \circ d$ define the same topology on $X$ for each ultrametric space $(X,d)$. The set of all strongly ultrametric preserving…

We prove consistency of the following sentence: ``ZFC + every real function is continuous on a non-meagre set'', answering a question of Fremlin.

Let $\mu$ be a positive measure on the real line with locally finite support $\Lambda$ and integer masses such that its Fourier transform in the sense of distributions is a purely point measure. An explicit form is found for an entire…

A Steinhaus set $S \subseteq \RR^d$ for a set $A \subseteq \RR^d$ is a set such that $S$ has exactly one point in common with $\tau A$, for every rigid motion $\tau$ of $\RR^d$. We show here that if $A$ is a finite set of at least two…

Assuming $\mathfrak b = \mathfrak c$ (or some weaker statement), we construct a compactification $\gamma\omega$ of $\omega$ such that its remainder $\gamma\omega\setminus\omega$ is nonseparable and carries a strictly positive measure.

A union ultrafilter is an ultrafilter over the finite subsets of $\omega$ that has a base of sets of the form $\mathrm{FU}(X)$, where $X$ is an infinite pairwise disjoint family and $\mathrm{FU}(X)=\{\bigcup…

We prove the existence of clopen marker sets with some strong regularity property. For each $n\geq 1$ and any integer $d\geq 1$, we show that there are a positive integer $D$ and a clopen marker set $M$ in $F(2^{\mathbb{Z}^n})$ such that…

By Zeckendorf's Theorem, every positive integer is uniquely written as a sum of non-adjacent terms of the Fibonacci sequence, and its converse states that if a sequence in the positive integers has this property, it must be the Fibonacci…

We introduce a notion of weak convergence in arbitrary metric spaces. Metric functionals are key in our analysis: weak convergence of sequences in a given metric space is tested against all the metric functionals defined on said space. When…

Rouch\'e's Theorem is among the most useful results in complex analysis for counting zeros of analytic functions. Rouch\'e's Theorem also admits a harmonic analogue for counting zeros of complex harmonic functions. Previously, this analogue…