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Boaz Tsaban

Publications and source records attributed to Boaz Tsaban.

At least 19 recordsLinked to original sources
arXiv2026

The double coset problem for parabolic subgroups of braid groups

We solve the double coset problem for arbitrary parabolic subgroups of the $n$-strand braid group $B_n$, by reducing it to the simultaneous conjugacy problem in the larger group $B_{n+1}$. The resulting decision algorithm is uniform in the braid index and the supplied subgroup generators. We also formulate the reduction for general group embeddings, and apply it to additional commuting constraints, several other spherical Artin types, and finite direct products.

math.GR↗
arXiv2026

Partition regularity of infinite parallelepiped sets

A proper infinite parallelepiped (IP) set in a semigroup is an infinite set consisting of a sequence $\myseq{a}$ and its finite sums, or a superset of such a set. Hindman's theorem asserts that the proper IP sets of natural numbers are partition regular: for each finite coloring of a proper IP set of natural numbers there is a monochromatic proper IP subset. Furstenberg generalized this question to arbitrary semigroups, in which the analogous result does not hold in general. We provide a complete classification of the semigroups for which the proper IP sets are partition regular, and show that this property is equivalent to other fundamental notions of additive Ramsey theory.

math.CO↗
arXiv2026

A solution of the D-space Problem

Settling a central problem in set-theoretic topology, and many related problems, we prove that there is, consistently, a regular Lindelöf space of cardinality $\aleph_1$ that is not a D-space. The existence of such a space is independent of ZFC; we establish its equivalence to a weak form of the Continuum Hypothesis. The proof of the main theorem is adjustable to produce consistent examples with a variety of stronger properties.

math.GN↗
arXiv2026

Combinatorial covering properties in an uncountable setting: canonical examples

We provide examples of spaces satisfying generalized combinatorial covering properties such as the Hurewicz, Menger, and $γ$-properties in an uncountable setting. Our approach is motivated by canonical constructions from the classical countable case, including the examples of Bartoszyński and Shelah separating the Hurewicz property from $σ$-compactness, the examples of Tsaban and Zdomskyy separating the Hurewicz and Menger properties, and Tsaban's construction of a nontrivial set of reals with the $γ$-property. We focus on the genuinely nontrivial aspects of these higher-cardinal generalizations, uncovering several open problems whose nature appears substantially different from that of their countable counterparts.

math.GN↗
arXiv2023

Selection principles and proofs from the Book

I provide simplified proofs for each of the following fundamental theorems regarding selection principles: 1. The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting that a certain, important property of the space of continuous functions on a space is actually preserved by Borel images of that space. 2. The Scheepers Diagram Last Theorem, due to Peng, completing all provable implications in the diagram. 3. The Menger Game Theorem, due to Telgársky, determining when Bob has a winning strategy in the game version of Menger's covering property. 4. A lower bound on the additivity of Rothberger's covering property, due to Carlson. The simplified proofs lead to several new results.

math.GN↗
arXiv2020

Finite powers and products of Menger sets

We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in the Blass--Shelah model for arbitrary values of the ultrafilter and dominating number.

math.GN↗
arXiv2019

Strongly sequentially separable function spaces, via selection principles

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.

math.GN↗
arXiv2019

Conceptual proofs of the Menger and Rothberger games

We provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewicz's Theorem concerning the Menger game, and Pawlikowski's Theorem concerning the Rothberger game.

math.GN↗
arXiv2018

Selective covering properties of product spaces, II: gamma spaces

We study productive properties of gamma spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: 1. Solving a problem of F. Jordan, we show that for every unbounded tower set of reals X of cardinality aleph_1, the space Cp(X) is productively FU. In particular, the set X is productively gamma. 2. Solving problems of Scheepers and Weiss, and proving a conjecture of Babinkostova-Scheepers, we prove that, assuming CH, there are gamma spaces whose product is not even Menger. 3. Solving a problem of Scheepers-Tall, we show that the properties gamma and Gerlits--Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that Remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems, and use Arhangel'skii duality to obtain results concerning local properties of function spaces and countable topological groups.

math.LO↗
arXiv2018

The Haar Measure Problem

An old problem asks whether every compact group has a Haar-nonmeasurable subgroup. A series of earlier results reduce the problem to infinite metrizable profinite groups. We provide a positive answer, assuming a weak, potentially provable, consequence of the Continuum Hypothesis. We also establish the dual, Baire category analogue of this result.

math.GN↗
arXiv2018

Complete simultaneous conjugacy invariants in Artin's braid groups

We solve the simultaneous conjugacy problem in Artin's braid groups and, more generally, in Garside groups, by means of a complete, effectively computable, finite invariant. This invariant generalizes the one-dimensional notion of super summit set to arbitrary dimensions. One key ingredient in our solution is the introduction of a provable high-dimensional version of the Birman--Ko--Lee cycling theorem. The complexity of this solution is a small degree polynomial in the cardinalities of our generalized super summit sets and the input parameters. Computer experiments suggest that the cardinality of this invariant, for a list of order $N$ independent elements of Artin's braid group $B_N$, is generically close to~1.

math.GR↗
arXiv2017

Menger's and Hurewicz's Problems: Solutions from "The Book" and refinements

We provide simplified solutions of Menger's and Hurewicz's problems and conjectures, concerning generalizations of sigma-compactness. The reader who is new to this field will find a self-contained treatment in Sections 1, 2, and 5. Sections 3 and 4 contain new results, based on the mentioned simplified solutions. The main new result is that there is a set of reals X of cardinality equal to the unbounding number b, and which has the following property: "Given point-cofinite covers U_1,U_2,... of X, there are for each n sets u_n,v_n in U_n, such that each member of X is contained in all but finitely many of the sets u_1 union v_1,u_2 union v_2,..." This property is strictly stronger than Hurewicz's covering property, and by a result of Miller and the present author, one cannot prove the same result if we are only allowed to pick one set from each U_n.

math.GN↗
arXiv2017

Products of general Menger spaces

We study products of general topological spaces with Menger's covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming \CH{}, every productively Lindelöf space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.

math.GN↗
arXiv2017

A classification of the cofinal structures of precompacta

We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups $\C(X,\bbR)$ of continuous real-valued functions on complete metric spaces $X$, with respect to the compact-open topology.

math.GN↗
arXiv2016

SPM Bulletin 40

This issue surveys some of the activities in the field since the previous issue, and announces a conference fully dedicated to the topic of selection principles.

math.GN↗
arXiv2016

SPM Bulletin 39

With the approaching TOPOSYM'16 (http://www.toposym.cz/programme.php), it is a pleasure to see selection principles gain increasing attention and becoming a standard part of topology and set theory. At least eight of the 28 speakers, and a good number of the contributed lecture speakers, made substantial contributions to this topic in their career. For some of these, SPs constitute the main topic of research in the last few years. This is in accordance with the continuous progress on the topic, some of which reported in this bulletin.

math.GN↗
arXiv2016

A Practical Cryptanalysis of the Algebraic Eraser

Anshel, Anshel, Goldfeld and Lemieaux introduced the Colored Burau Key Agreement Protocol (CBKAP) as the concrete instantiation of their Algebraic Eraser scheme. This scheme, based on techniques from permutation groups, matrix groups and braid groups, is designed for lightweight environments such as RFID tags and other IoT applications. It is proposed as an underlying technology for ISO/IEC 29167-20. SecureRF, the company owning the trademark Algebraic Eraser, has presented the scheme to the IRTF with a view towards standardisation. We present a novel cryptanalysis of this scheme. For parameter sizes corresponding to claimed 128-bit security, our implementation recovers the shared key using less than 8 CPU hours, and less than 64MB of memory.

math.GR↗