Scale pressure for amenable group actions

The paper’s Theorem A states and proves that for amenable actions satisfying Condition A and along tempered Følner sequences with |F_n|/log n → ∞, one has SP(ϕ,s) = sp(ϕ,s,δ) for every δ ∈ (0,1) (see Theorem A and surrounding definitions and lemmas in the uploaded PDF: Condition A and sp are defined and Theorem A stated on pp. 7–9; proof steps are given on pp. 10–12 ). The model’s solution outlines a broadly similar comparison strategy, but it contains a critical error: it attempts to control the measure of ⋃_{g∈F_n} g^{-1}U_r(ξ) by the crude union bound |F_n|·µ(U_r(ξ)), and then claims this can be made small by taking n large. This is backwards—the bound grows with |F_n|—and the paper avoids this pitfall via a precise combinatorial covering estimate (the D(·) term) to relate Bowen balls to partition atoms, together with Zhao’s entropy formula h_µ(G) = lim_{ε→0} h_{µ,δ,ε}(X,G) (uniform in fixed δ) along such Følner sequences (definitions and Proposition 2.3: pp. 4–7; Theorem A proof uses these tools on pp. 10–12 ). Hence, the paper’s argument is correct as written, while the model’s proof has a substantive gap.