Abelian covers of hyperbolic surfaces: equidistribution of spectra and infinite mixing asymptotics for horocycle flows

The paper proves a full asymptotic expansion for horocycle correlations on Zd-covers (Theorem A), with leading constant ((g−1)/2)^{d/2} σ(Γ0) and a complete series in powers of (log t)−1; the proof proceeds via a character (Fourier) decomposition along the deck group, precise asymptotics of twisted matrix coefficients, and a multidimensional stationary phase/Laplace method in the large parameter T = log t. All these ingredients, including the spectral reduction near the trivial representation, Proposition 3.6’s Hessian identification, Lemma 4.1’s stationary-phase constant (2π)^{d/2}, and the final evaluation A0 = vol(v)vol(w), are explicitly established in the paper. The candidate solution mirrors the paper’s strategy and arrives at the same final expansion and leading constant. However, it misstates two technical points: (i) it writes the time-exponent as e^{−(log t)·λ1(ξ)} instead of the paper’s t−1+ν0(ξ) = e^{(log t)(−1+ν0(ξ))}; and (ii) it omits the factor 2π in the Hessian normalization of λ0 at 0. These slips cancel in the end so that the leading constant still matches Theorem A, but the intermediate statements should be corrected to align with the paper’s precise normalization and exponent. Overall, the model’s argument is the same in substance as the paper’s and reaches the correct conclusion, with minor normalization and exponent bookkeeping errors. See the paper’s statement of Theorem A and its proof via Lemma 3.3, Theorem 2.2, Proposition 3.6, Lemma 4.1, and equation (18) leading to the final constant and series .