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Nov 5, 2023 at 20:20 comment added Yemon Choi @DavidGao That's a good question. Indeed I haven't thought about $\{\pm 1\}^{\oplus \mathbb N}$ where an answer - positive or negative - must surely be known via Fourier analysis on LCA groups
Nov 5, 2023 at 18:11 comment added David Gao Thanks! I suppose you meant something like, in the definition of $f_n$ we can replace $g$ by any group element of order larger than $n$, then the proof basically works as is for infinite torsion groups with unbounded orders of elements? For the remainder case, do you know if this holds in some specific cases, such as the direct sum or Cartesian product of infinite copies of a fixed finite group?
Nov 5, 2023 at 14:26 comment added Yemon Choi On further reflection, I think that this should also yield counterexamples for any $G$ where the orders of elements can be arbitrarily large (just use the RS polynomials but restrict to finite cyclic subgroups). Thus the problem is reduced to infinite torsion groups with a uniform bound on the orders of elements
Nov 5, 2023 at 14:04 history answered Yemon Choi CC BY-SA 4.0