Shrinking Target Horospherical Equidistribution via Translated Farey Sequences

The paper proves the spherical shrinking-target equidistribution (Theorem 1.5) with the precise normalization T^{d-1} and uniformity for T in [T0, e^{η t}] using a renormalization strategy via (translated) Farey sequences and detailed volume/disjointness computations in Sections 4–6; the constant is identified using an exact volume scaling (Theorem 5.10 shows T^{d-1} μ(ST_E) is independent of T) . The candidate's solution outlines a different, standard approach: smoothing the target in G, applying effective mixing/decay of matrix coefficients to horospherical averages, and exploiting the same T^{-(d-1)} scaling, yielding the same limit and uniformity. While the model compresses several technical steps (wavefront control in the cusp, Sobolev norm bounds under smoothing, and an explicit effective equidistribution inequality) into citations, these ingredients are known in the literature and align with the statement of the theorem. The paper’s proof is complete and rigorous via its own machinery, and the model’s proof sketch is correct in outline but would need the stated technical tools to be spelled out for full rigor.