QUANTITATIVE CONVERGENCE FOR SPARSE ERGODIC AVERAGES IN L1

The paper proves uniform weak-(1,1) jump/variation/oscillation bounds along lacunary times for the sequences a_n = ⌊n^c⌋ with 1<c<8/7 and for random hitting times with EX_n=n^{-α}, 0<α<1/2 (Theorem 1.4), via a calibrated Calderón–Zygmund scheme anchored by three structural hypotheses (ℓ2-boundedness, sparse support, reflection symmetry) and a detailed verification for the deterministic and random models (Proposition 3.2 and Section 4) . The candidate solution outlines a different proof strategy based on a lacunary square function of increments S_D and an asserted L^2 multiplier difference bound sup_θ ∑_j |m_{N_{j+1}}−m_{N_j}|^2<∞, but it neither proves this key bound nor is it the route taken in the paper; it also treats as “pointwise” an inequality with the (r−2)/r factor that is known in operator/norm form, not as a general pointwise comparison. The model therefore hinges on unproven steps and a mis-citation that the paper contains the required S_D argument (it does not).