Productivity of sequences with respect to a given weight function

Abstract

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: (i) f-Cauchy productive ( f-productive) provided that the sequence { ∏ n = 0 m a n z ( n ) : m ∈ N } is left Cauchy (converges to some element of G, respectively) for each function z : N → Z such that | z ( n ) | ⩽ f ( n ) for every n ∈ N ; (ii) unconditionally f-Cauchy productive ( unconditionally f-productive) provided that the sequence { a φ ( n ) : n ∈ N } is ( f ∘ φ ) -Cauchy productive (respectively, ( f ∘ φ ) -productive) for every bijection φ : N → N . (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 { ∏ n = 0 m 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 C p ( − , G ) -theory.