Productivity of sequences with respect to a given weight function

AbstractGiven a function f:N→(ω+1)∖{0}, we say that a faithfully indexed sequence {an:n∈N} of elements of a topological group G is: (i) f-Cauchy productive (f-productive) provided that the sequence {∏n=0manz(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 {aφ(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 fω-Cauchy productive sequence, where fω is the function taking the constant value ω. We give an example of an fω-productive sequence {an: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=0maφ(n):m∈N} diverges, thereby answering a question of Dominguez and Tarieladze. Furthermore, we show that H has no unconditionally fω-productive sequences. As an application of our results, we resolve negatively a question from Cp(−,G)-theory