TRIMMED ERGODIC SUMS FOR NON-INTEGRABLE FUNCTIONS WITH POWER SINGULARITIES OVER IRRATIONAL ROTATIONS

The paper proves, in full detail, that for f(x)=x^{-β} with β>1 and a monotone trimming sequence k(N)=o(N), the following are equivalent: a uniform trimmed strong law, a trimmed weak law, and the arithmetic condition k(N)/max(b_n, max_{j≤n−1} a_j)→∞, with the normalization d_N=(β−1)^{-1}N^β k(N)^{1−β} (Theorem 12). The candidate solution states exactly this theorem and gives a different proof outline based on a uniform control of order statistics via the three-gap/Sturmian structure, plus a Riemann-sum comparison and a cluster argument for necessity. The paper’s proof is rigorous and complete; the model’s approach is plausible but leaves its key uniform order-statistic lemma as a sketch. Hence both are correct, using different proof strategies, with the paper providing full details.