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Productivity of Sequences with Respect to a Given Weight Function by Dikran Dikranjan; Dmitri Shakhmatov; Jan Spěvák is a Mathematics article available to read on EtoBox.

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Given a function f : N → (ω + 1) \{0}, we say that a faithfully indexed sequence {a n : n ∈ N} of elements of a topological group G is: (Bijections can be replaced by injections here.) We consider the question of existence of (unconditionally) f -productive sequences for a given "weight function" f . We prove that: (1) a Hausdorff group having an f -productive sequence for some f contains a homeomorphic copy of the Cantor set; (2) if a non-discrete group is either locally compact Hausdorff or Weil complete metric, then it contains an unconditionally f -productive sequence for every function f : N → N \ {0}; (3) a metric group is NSS if and only if it does not contain an f ω -Cauchy productive sequence, where f ω is the function taking the constant value ω. We give an example of an f ω -productive sequence {a n : n ∈ N} in a (necessarily non-abelian) separable metric group H with a linear topology and a bijection φ : N → N such that the sequence { m n=0 a φ(n) : m ∈ N} diverges, thereby answering a question of Dominguez and Tarieladze. Furthermore, we show that H has no unconditionally f ω -productive sequences. As an application of our results, we resolve negatively a question from

Who reads Productivity of Sequences with Respect to a Given Weight Function?

Author

Dikran Dikranjan; Dmitri Shakhmatov; Jan Spěvák

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0166-8641)

Published

2011

Language

EN

Field

Mathematics (Physical Sciences)