PRECISE ASYMPTOTICS ON THE BIRKHOFF SUMS FOR DYNAMICAL SYSTEMS

The paper proves, under CLT and a refined LD assumption, that ε^2 Σ_n Λ_n(ε) → σ^2 and (−log ε)^{-1} Σ_n Λ_n(ε)/n → 2. The candidate solution establishes the same limits by the same bulk/tail split at n≈ε^{-2}, a uniform CLT (via Polya) on compact t-intervals, Gaussian approximation in the bulk, Riemann-sum limits, and LD control of the tail. This matches the paper’s approach (which uses Euler–Maclaurin to evaluate the Gaussian sum precisely) and assumptions. Minor technical differences (Riemann-sum vs. Euler–Maclaurin; treating ± simultaneously vs. splitting) do not affect correctness. Thus both are correct and substantially the same.