Stable functions and Følner’s Theorem

The paper proves the Bohr0–inside–AA^{-1} up to BD-null result for all amenable groups via weakly almost periodic flows, the Ellis semigroup, and unique invariant measure, with a clear, complete argument. The candidate solution sketches a different representation-theoretic route (GNS + Jacobs–de Leeuw–Glicksberg), but its crucial step—that the weakly-mixing component’s deviation set N_ε = {g : |φ_0(g)|>ε} has upper Banach density 0—is asserted without justification and, as stated, does not follow from mean-zero alone; in fact, mean zero does not imply BD-null of {|f|>ε}. Moreover, that step is cited back to the same paper, rendering the purportedly different proof circular. Hence: paper correct; model (as written) not correct/complete.