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Boaz Tsaban

Publications and source records attributed to Boaz Tsaban.

At least 37 records · Page 2Linked to original sources
arXiv2016

Products of Menger spaces: a combinatorial approach

We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in all but the most exotic models of real set theory. On the other hand, we establish productive properties for versions of Menger's property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger's property is strictly between Menger's and Hurewicz's classic properties. We include a number of open problems emerging from this study.

math.GN↗
arXiv2016

Arhangel'ski\uı sheaf amalgamations in topological groups

We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos's property $α_{1.5}$ is equivalent to Arhangel'ski\uı's formally stronger property $α_1$. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space $X$ such that the space $C_p(X)$ of continuous real-valued functions on $X$, with the topology of pointwise convergence, has Arhangel'ski\uı's property $α_1$ but is not countably tight. This result follows from results of Arhangel'ski\uı--Pytkeev, Moore and Todorčević, and provides a new solution, with remarkable properties, to a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces. The Averbukh--Smolyanov problem was first solved by Plichko (2009), using Banach spaces with weaker locally convex topologies.

math.GN↗
arXiv2015

SL2 homomorphic hash functions: Worst case to average case reduction and short collision search

We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the field with $q$ elements. Modulo a well supported number theoretic hypothesis, which holds in particular for concrete homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions: upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism. For a family of homomorphisms containing several concrete proposals in the literature, we prove that collisions of length O(log(q)) can be found in running time O(sqrt(q)). For general homomorphisms we offer an algorithm that, heuristically and according to experiments, in running time O(sqrt(q)) finds collisions of length O(log(q)) for q even, and length O(log^2(q)/loglog(q))$ for arbitrary q. While exponetial time, our algorithms are faster in practice than all earlier generic algorithms, and produce much shorter collisions.

cs.CR↗
arXiv2015

SPM Bulletin 38

In addition to research announcements, this issue features a call for papers to a Topology and its Applications special issue, and an intriguing open problem.

math.GN↗
arXiv2015

Algebra, selections, and additive Ramsey theory

Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods include, in addition to Hurewicz's game theoretic characterization of Menger's property, extensions of the classic idempotent theory in the Stone--Czech compactification of semigroups, and of the more recent theory of selection principles. This provides strong versions of the mentioned celebrated theorems, where the monochromatic substructures are large, beyond infinitude, in an analytic sense. Reducing the main theorems to the purely combinatorial setting, we obtain nontrivial consequences concerning uncountable cardinal characteristics of the continuum. The main results, modulo technical refinements, are of the following type (definitions provided in the main text): Let $X$ be a Menger space, and $\mathcal{U}$ be an infinite open cover of $X$. Consider the complete graph, whose vertices are the open sets in $X$. For each finite coloring of the vertices and edges of this graph, there are disjoint finite subsets $\mathcal{F}_1,\mathcal{F}_2,\dots$ of the cover $\mathcal{U}$ whose unions $V_1 := \bigcup\mathcal{F}_1, V_2 := \bigcup\mathcal{F}_2,\dots$ have the following properties: 1. The sets $\bigcup_{n\in F}V_n$ and $\bigcup_{n\in H}V_n$ are distinct for all nonempty finite sets $F<H$. 2. All vertices $\bigcup_{n\in F}V_n$, for nonempty finite sets $F$, are of the same color. 3. All edges $\bigl\{\,\bigcup_{n\in F}V_n, \bigcup_{n\in H}V_n\,\bigr\}$, for nonempty finite sets $F<H$, have the same color. 4. The family $\{V_1,V_2,\dots\}$ is an open cover of $X$. A self-contained introduction to the necessary parts of the needed theories is provided.

math.GN↗
arXiv2015

Combinatorial aspects of selective star covering properties in $Ψ$-spaces

Which Isbell--Mrówka spaces ($Ψ$-spaces) satisfy the star version of Menger's and Hurewicz's covering properties? Following Bonanzinga and Matveev, this question is considered here from a combinatorial point of view. An example of a $Ψ$-space that is (strongly) star-Menger but not star-Hurewicz is obtained. The PCF-theory function $κ\mapsto\cof([κ]^\alephes)$ is a key tool. Using the method of forcing, a complete answer to a question of Bonanzinga and Matveev is provided. The results also apply to the mentioned covering properties in the realm of Pixley--Roy spaces, to the extent of spaces with these properties, and to the character of free abelian topological groups over hemicompact $k$ spaces.

math.GN↗
arXiv2015

Polynomial-time solutions of computational problems in noncommutative-algebraic cryptography

We introduce the \emph{linear centralizer method}, and use it to devise a provable polynomial time solution of the Commutator Key Exchange Problem, the computational problem on which, in the passive adversary model, the security of the Anshel--Anshel--Goldfeld 1999 \emph{Commutator} key exchange protocol is based. We also apply this method to the computational problem underlying the \emph{Centralizer} key exchange protocol, introduced by Shpilrain and Ushakov in 2006. This is the first provable polynomial time cryptanalysis of the Commutator key exchange protocol, hitherto the most important key exchange protocol in the realm of noncommutative-algebraic cryptography, and the first cryptanalysis (of any kind) of the Centralizer key exchange protocol. Unlike earlier cryptanalyses of the Commutator key exchange protocol, our cryptanalyses cannot be foiled by changing the distributions used in the protocol.

cs.CR↗
arXiv2014

SPM Bulletin 37

In the year 2014, the field of selection principles found its way into several additional, fascinating mathematical realms. The field enters the consensus as a mainstream part of set theory and topology, and as a promising direction for young researchers that are well trained in these fields. Some of the interesting developments in the field during 2014 are reported in this issue.

math.GN↗
arXiv2014

The character of topological groups, via bounded systems, Pontryagin--van Kampen duality and pcf theory

The Birkhoff--Kakutani Theorem asserts that a topological group is metrizable if and only if it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups. We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are the closed subgroups of Pontryagin--van Kampen duals of \emph{metrizable} abelian groups, or equivalently, complete abelian groups whose dual is metrizable. By investigating these connections, we show that also in these cases, the character can be estimated, and that it is determined by the weights of the \emph{compact} subsets of the group, or of quotients of the group by compact subgroups. It follows, for example, that the density and the local density of an abelian metrizable group determine the character of its dual group. Our main result applies to the more general case of closed subgroups of Pontryagin--van Kampen duals of abelian Čech-complete groups. In the special case of free abelian topological groups, our results extend a number of results of Nickolas and Tkachenko, which were proved using combinatorial methods. In order to obtain concrete estimations, we establish a natural bridge between the studied concepts and pcf theory, that allows the direct application of several major results from that theory. We include an introduction to these results and their use.

math.FA↗
arXiv2014

The linear refinement number and selection theory

The \emph{linear refinement number} $\mathfrak{lr}$ is the minimal cardinality of a centered family in $[ω]^ω$ such that no linearly ordered set in $([ω]^ω,\subseteq^*)$ refines this family. The \emph{linear excluded middle number} $\mathfrak{lx}$ is a variation of $\mathfrak{lr}$. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classic combinatorial cardinal characteristics of the continuum. We prove that $\mathfrak{lr}=\mathfrak{lx}=\mathfrak{fd}$ in all models where the continuum is at most $\aleph_2$, and that the cofinality of $\mathfrak{lr}$ is uncountable. Using the method of forcing, we show that $\mathfrak{lr}$ and $\mathfrak{lx}$ are not provably equal to $\mathfrak{d}$, and rule out several potential bounds on these numbers. Our results solve a number of open problems.

math.GN↗
arXiv2013

SPM Bulletin 36

In addition to a number of new developments in the field, this issue announces the completion of the special issue of Topology and its Applications, dedicated to the proceedings of the Fourth Workshop on Coverings, Selections and Games in Topology, on the occasion of Ljubiša D.R. Kočinac's 65th birthday.

math.GN↗
arXiv2013

A reduction of semigroup DLP to classic DLP

We present a polynomial-time reduction of the discrete logarithm problem in any periodic (a.k.a. torsion) semigroup (SGDLP) to the same problem in a subgroup of the same semigroup. It follows that SGDLP can be solved in polynomial time by quantum computers, and that SGDLP has subexponential algorithms whenever the classic DLP in the corresponding groups has subexponential algorithms.

cs.CR↗
arXiv2013

Hindman's Coloring Theorem in arbitrary semigroups

Hindman's Theorem asserts that, for each finite coloring of the natural numbers, there are distinct natural numbers $a_1,a_2,\dots$ such that all of the sums $a_{i_1}+a_{i_2}+\dots+a_{i_m}$ ($m\ge 1$, $i_1<i_2<\dots<i_m$) have the same color. The celebrated Galvin--Glazer proof of Hindman's Theorem and a classification of semigroups due to Shevrin, imply together that, for each finite coloring of each infinite semigroup $S$, there are distinct elements $a_1,a_2,\dots$ of $S$ such that all but finitely many of the products $a_{i_1}a_{i_2}\cdots a_{i_m}$ ($m\ge 1$, $i_1<i_2<\dots<i_m$) have the same color. Using these methods, we characterize the semigroups $S$ such that, for each finite coloring of $S$, there is an infinite \emph{subsemigroup} $T$ of $S$, such that all but finitely many members of $T$ have the same color. Our characterization connects our study to a classical problem of Milliken, Burnside groups and Tarski Monsters. We also present an application of Ramsey's graph-coloring theorem to Shevrin's theory.

math.CO↗
arXiv2013

SPM Bulletin 35

This issue contains a number of abstracts of papers directly related to selection principles, some of which presented at the conference SPMC2012, and the announcement of the solution of Malyhin's Problem.

math.GN↗
arXiv2013

Selective covering properties of product spaces

We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are (definitions provided in the paper): \be \item Every product of a concentrated space with a Hurewicz $\sone(\Ga,\Op)$ space satisfies $\sone(\Ga,\Op)$. On the other hand, assuming \CH{}, for each Sierpiński set $S$ there is a Luzin set $L$ such that $L\x S$ can be mapped onto the real line by a Borel function. \item Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. \item Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits--Nagy. \item Assuming $\fd=\aleph_1$, every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers. \ee A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than $\add(\cN)$, the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy with $\add(\cN)<\cov(\cM)$. Our results improve upon and unify a number of results, established earlier by many authors.

math.GN↗
arXiv2012

Monochromatic generating sets in groups and other algebraic structures

The \emph{generating chromatic number} of a group $G$, $\chigen(G)$, is the maximum number of colors $k$ such that there is a monochromatic generating set for each coloring of the elements of $G$ in $k$ colors. If no such maximal $k$ exists, we set $\chigen(G)=\infty$. Equivalently, $\chigen(G)$ is the maximal number $k$ such that there is no cover of $G$ by proper subgroups ($\infty$ if there is no such maximal $k$). We provide characterizations, for arbitrary gruops, in the cases $\chigen(G)=\infty$ and $\chigen(G)=2$. For nilpotent groups (in particular, for abelian ones), all possible chromatic numbers are characterized. Examples show that the characterization for nilpotent groups do not generalize to arbitrary solvable groups. We conclude with applications to vector spaces and fields.

math.GR↗
arXiv2012

Pointwise convergence of partial functions: The Gerlits-Nagy Problem

For a set $X\sbst\R$, let $B(X)\sbst\R^X$ denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product $\R^X$. Assume that for each countable $A\sbst B(X)$, each $f$ in the closure of $A$ is in the closure of $A$ under pointwise limits of sequences of partial functions. We show that in this case, $B(X)$ is countably Fréchet--Urysohn, that is, each point in the closure of a countable set is a limit of a sequence of elements of that set. This solves a problem of Arnold Miller. The continuous version of this problem is equivalent to a notorious open problem of Gerlits and Nagy. Answering a question of Salvador Hernańdez, we show that the same result holds for the space of all Baire class 1 functions on $X$. We conjecture that, in the general context, the answer to the continuous version of this problem is negative, but we identify a nontrivial context where the problem has a positive solution. The proofs establish new local-to-global correspondences, and use methods of infinite-combinatorial topology, including a new fusion result of Francis Jordan.

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