On asymptotic expansions of ergodic integrals for Zd-extensions of translation flows.

The paper establishes asymptotic expansions for ergodic integrals of translation flows on Zd-covers under (H1)–(H3), using renormalization by a pseudo-Anosov map, a fiber-Fourier/Td-twist, analytic perturbation of twisted transfer operators Lu near u=0, and inverse Fourier analysis. This is exactly the spectral/Nagaev–Guivarc’h strategy articulated in the candidate solution. Concretely: (i) the renormalization identity Ax,T(f)=AψΓ^k(x),λ^kT(f∘ψΓ^{-k}) is used to reduce long trajectories to short ones (Lemma 4.5) ; (ii) deck-Fourier transforms and the twisted operator Lu are introduced, giving an exact operator/Fourier representation (Lemma 2.3) ; (iii) on suitable anisotropic Banach spaces, L has a spectral gap and Lu depends analytically on u, yielding L^n_u=λ(u)^nΠ(u)+Q(u)^n with ∥Q(u)^n∥≤δ0^n (Proposition 2.4) and a local Gaussian expansion for λ(u) (20) ; (iv) this produces an operator LLT with expansions (Proposition 2.5 and Corollary 2.7) and the main asymptotics (Theorems 1.1, 2.8), including the explicit d=1 formula and the d=2 analogue up to displayed orders . The amplitude normalization used by the model, namely F(0)=∫XΓ f dm, coincides with the paper’s identity (28) derived from renormalization and invariance (Lemma 4.5 and the definition of F(α)) . The paper’s technical construction that the observable on short segments has uniform smoothness in the twist and yields bounded coefficients matches the model’s H_{u,K} device via partition refinements PR and the fact that the Frobenius cocycle F is locally constant (Lemma 2.18) . As for the resulting expansions, the model’s d=1 formula (T/√K, Gaussian factor, polynomial Edgeworth-type corrections in K^{-1}) matches Theorem 1.1 precisely; and for d=2 the model captures the same structure (Gaussian kernel in ξ(ψΓ^K(x))/√K times T/K with bounded coefficient functions), consistent with the proof of Theorem 2.8 and the displayed d=2 expansion fragment in the paper (up to linear terms and higher-order O(T/(√K)^5) remainder as shown in the text) . Minor stylistic differences remain: the paper works on anisotropic Banach spaces for the invertible map ψ and gives two complementary constructions, whereas the model sketches an equivalent coding/Ruelle approach; both routes implement the same Nagaev–Guivarc’h spectral-perturbative mechanism. Overall, the logical steps, hypotheses, and normalization identities concur.