Building prescribed orbit equivalence: Sofic approximations and optimal quantitative orbit equivalence

The paper proves the statement (Theorem 1.16) that for any ρ in the class C with Σm ρ(κ^m) κ^{-m} < ∞, there is a finitely generated group G with IG ≃ ρ∘log and a (ρ∘log, L0)-integrable orbit-equivalence coupling to Z, using an explicit sofic-approximation construction and a quantitative criterion (eq. (2.1)) to verify integrability . The model’s solution claims to achieve the same integrability via Følner tiling couplings by tuning tile diameters and errors; however, the paper explicitly notes that the Følner tiling approach “does not seem to produce couplings with optimal integrability,” yielding only a logarithmic loss (e.g., Theorem 1.15) rather than the optimal ρ∘log achieved here using sofic approximations . The existence of G with IG ≃ ρ∘log is correctly realized via Brieussel–Zheng diagonal products (Appendix A), which both the paper and the model agree on . But the model’s assertion that the optimal integrability follows from Følner tiling shifts under the stated summability is unsupported and contradicts the paper’s methodological assessment.