p-adic Directions of Primitive Vectors

The paper proves joint equidistribution of real and p-adic directions of primitive integer vectors with an effective power-saving error via S-arithmetic lattice point counting for Lipschitz well-rounded sets and Gorodnik–Nevo’s quantitative equidistribution. Theorem 1 states the limit for arcs Θ×Θp with error O(R^{-2τp+δ}), τp=1/28 unconditionally, improving to 1/14 under Ramanujan, and sets up the bijection with Γ=SL2(Z[1/p]) using Bruhat–Iwasawa coordinates (Proposition 13), LWR at the p-adic place (Proposition 20), and the counting theorem (Theorem 17) . The candidate solution follows the same blueprint: it works on Γ\G with G=SL2(R)×SL2(Qp), encodes primitive vectors via column maps, builds product LWR families encoding real and p-adic arcs, applies Gorodnik–Nevo for the main term and error, and derives the same exponent τp, including the Ramanujan improvement. Minor differences are cosmetic: (i) the paper’s correspondence uses the first column v⊥ rather than v itself, but this is immaterial for measuring directions; (ii) the paper explicitly uses the right half p-adic unit sphere S1,p^+ (Fact 11) in intermediate ratios, then passes to the full S1p when forming the final normalization, whereas the candidate discusses normalization directly. Both arguments are consistent on measure conventions (Definition 5; µ2p(θ+p^N Zp^2)=p^{-2N}) and on the error exponent coming from decay of matrix coefficients (Remark 18). Overall, the proofs are the same in method and substance.