THE DISTRIBUTION OF DILATING SETS: A JOURNEY FROM EUCLIDEAN TO HYPERBOLIC GEOMETRY

The paper’s Theorem 5.2 explicitly states the same asymptotic expansion (including the cos/sin oscillatory terms at frequency r_λ, the special λ = 1/4 term with te^{-t/2}, the b ≠ 0 hypothesis for negative times, and a uniform 1/σ gain in both coefficients and remainder) that the candidate solution derives, with the same Sobolev threshold s > 11/2 and remainder O((t+1)e^{-t})/σ; see the theorem as displayed in (5.6) and the surrounding discussion , and Corollary 5.4 for the leading-order rate . The paper’s proof approach is representation-theoretic and Ratner-based (Section 5 and the proof outline in Section 6) rather than via Ruelle–Pollicott spectral theory . The candidate solution invokes the Ruelle spectrum and its identification with Laplace eigenvalues on compact hyperbolic surfaces, which also yields exactly the same expansion and rates. The only caveat is that the model sketches, but does not fully justify, the O(1/σ) dual-norm bound for the short W-segment in the specific anisotropic spaces; this is plausible and standard in spirit but would require a precise choice of anisotropic norms. Overall, the results match and the proofs are different in method.