Productivity of sequences with respect to a given weight function

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 {\prod_{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)|\leq 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 fω-Cauchy productive sequence, where f_ω is the function taking the constant value ω. We give an example of an f_ω-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 {\prod_{n=0}^ma^{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.