A UNIVERSAL LOWER BOUND FOR THE DISCREPANCIES OF ACTIONS OF A LOCALLY COMPACT GROUP

The paper proves the universal lower bound ||λ_G(f)||_{2→2} ≤ δ(f) under the explicit (S, F)-moderate growth hypothesis and supplies a complete, quantitative proof via a construction of test functions from B_n^± sets, convolution powers f^{*n}, and the Berg–Christensen asymptotic formula for μ_{f^{*n}}(F)^{1/n} = ||λ_G(f)||, see Theorem 1, its proof, and Lemma 1 in the paper. The candidate’s transference proof sketches a different route using matrix coefficients r_n and an intertwining operator L_n, but it relies on unproven steps: (i) a comparison “||λ_G(μ)|| ≤ liminf_n ||λ_G(μ * r_n)||”, (ii) the claim “||λ_G(r_n)|| → 1”, and (iii) an appeal to absent items (“Definition 6.1”) not present in the paper. These gaps are nontrivial and not justified by the cited hypotheses. Consequently, while the claimed inequality matches the paper’s main result, the model’s proof is incomplete/incorrect, whereas the paper’s argument is correct and self-contained .