Weighted Birkhoff averages: Deterministic and probabilistic perspectives

The paper proves the full Theorem 1.2 using Fourier expansion, Poisson summation, and an m-times integration-by-parts optimization that exploits sharp growth bounds on D^m w_{p,q} (via β_{p,q}=1+1/min{p,q}) to obtain the exp(-c N^{1/(τ+β_{p,q})}) mechanism, then couples this with Diophantine lower bounds and, in T_∞, lattice counting to get the stated rates, including the periodic case (III)-(7) with constants depending only on T and sup|f|. The candidate solution follows the same overall reduction to a kernel K_N(t), but swaps the paper’s integration-by-parts step for a Gevrey Paley–Wiener approach to get |K_N(t)| ≲ exp(-c (N||t||)^ζ) with ζ=(1+1/min{p,q})^{-1}, then reproduces the same rate optimizations, including the periodic case via rational t=ℓ/T. The exponents and quantifiers match Theorem 1.2 (items (I)(1–4), (II)(5–6), (III)(7)). Differences are methodological, not substantive, and no contradictions were found. The paper’s statements, scope, and constants are consistent with the candidate’s bounds. Citations: theorem statement and proof structure (e.g., Theorem 1.2 and Section 3.2) are explicit in the PDF ; the periodic-case Poisson step is shown in (3.10)–(3.11) ; the optimization m_N∼N^{1/(τ+β_{p,q})} and resulting exponent appear in (3.17)–(3.20) ; the infinite-dimensional cardinality/Diophantine handling (yielding exp(-(log N)^{ζ_3})) is in (3.23)–(3.30) .