FLUCTUATION BOUNDS FOR ERGODIC AVERAGES OF AMENABLE GROUPS

Warren’s Theorem 21 cleanly proves a bound of floor(||x||/(1/2·u(ε)−η)) ε-fluctuations at distance β(n, η/(3||x||)) (with the scaled variants and the sharpened version using a lower bound L) by combining a quantitative almost-invariance lemma (Lemma 19) with a two-step “norm drop” that avoids any continuity assumptions on the modulus u; see the statement and proof sketch around Theorem 21 and its setup , together with Lemma 19 and the preceding quantitative argument . The candidate solution largely mirrors this structure, but its Step 3 asserts a midpoint estimate that effectively replaces u(ε−η) by about (1/2)u(ε) by “absorbing” u(ε)−u(ε−η) into η; this absorption is unjustified without further regularity of u and, moreover, the subsequent use of ‖y‖ ≤ ‖(y+z)/2‖ + (1/2)‖y−z‖ treats a term for which only a lower bound is available as if it were small. The paper’s proof avoids both issues by introducing an intermediate index K and bounding AKx directly, yielding the correct constants and bound .