JOINT EQUIDISTRIBUTION ON THE PRODUCT OF THE CIRCLE AND THE UNIT TANGENT BUNDLE OF THE MODULAR SURFACE

The paper proves an O(n^{-1/2+δ}) joint equidistribution error term (any δ>7/64) for the primitive rational points paired with expanding horocycles on T×Γ\G via a clean spectral expansion: Fourier in the torus variable and Whittaker/Fourier expansions for cusp/Eisenstein spectra, with the 7/64 coming only from the known bound toward Ramanujan for GL(2) at finite places; see Theorem 1 and the outlined strategy (Fourier in T, spectral in Γ\G) together with Lemmas 2–3 and the torus-average estimate (equations (9)–(11), (13), (18)) in the PDF . By contrast, the candidate solution’s key Step (2) incorrectly identifies the averaging over k (with (k,n)=1) of the horocycle points as a Hecke-type spherical convolution operator H_n acting diagonally on Hecke eigenfunctions with Satake-controlled eigenvalues. The points Γ g_{n,k} in this problem are generated by unipotent translations along a horocycle and not by the standard Hecke double cosets, and the paper’s treatment does not use (nor seems to admit) such a Hecke-operator interpretation. The candidate also claims an O(n^{-1/2+ε}) bound for all nonzero Fourier modes via Kuznetsov/Weil, but for fixed cusp forms the paper’s sharp bound remains O(n^{-1/2+7/64+ε}), reflecting the necessity of Ramanujan-type input (Lemma 2), while O(n^{-1/2+ε}) is only available for the Eisenstein part (Lemma 3) . Hence the model’s conclusion matches the paper’s theorem numerically, but its central operator-theoretic step is not justified by the geometry of the averaging and its Kuznetsov claim is over-optimistic for fixed cuspidal input.