EFFECTIVE EQUIDISTRIBUTION OF PRIMITIVE RATIONAL POINTS ON EXPANDING HOROSPHERES

The paper proves an effective equidistribution theorem with two independent power-savings, exactly quantified in Theorem 1.2: Aq(f) equals the product measure integral with error O(S∞,κ+ε(f) q^{-ϑ+ε} + S2,κ′+ε(f) q^{-ϑ′+ε}) and with the piecewise values of ϑ, ϑ′, κ, κ′ stated there , built on the precise setup of G, H+, H, D(y), the rational points, and the averaging Aq(f) . The proof decomposes the main sum into E0, E1, E2 (eq. 7.19), handles E0 via a Hecke operator on SLd(Z)\SLd(R) with an explicit spectral bound (Prop. 4.2) , bounds E1 by a divisor-sum/geometry-of-numbers argument (eqs. 7.27–7.34) , and controls E2 using new bounds for matrix Kloosterman sums plus a geometric majorant and an averaging identity for the dual Hecke operator (eqs. 5.33–5.34 and 7.40–7.50) . The candidate solution gets the right exponents and norm-structure at a high level, but its key steps are incorrect or unjustified: (i) it asserts that {Γ hR} over primitive R is a single (Γ∩H)-double coset in H and builds a Hecke-type operator directly on Γ\ΓH; in the paper the Hecke operator acts on SLd(Z)\SLd(R) via the double coset for Dq (Lemma 4.1 and Prop. 4.2), with the primitive R parametrized by Bq×GLn(Z/qZ), not by a single H-double coset (Lemma 2.2) . (ii) It claims the normalized averaging operator "kills constants"; in fact, the normalized Hecke operator fixes constants and only contracts the mean-zero subspace (explicitly quantified in Prop. 4.2) . (iii) For the torus-error, it replaces the paper’s matrix Kloosterman-sum and geometric analysis by a brief Möbius-inversion heuristic; the paper’s E2 term shows that the limiting exponent ϑ=n−1 (or 1/2 for n=1) requires nontrivial Kloosterman-sum bounds and a careful argument with a dual Hecke average and Rogers’ formula (eqs. 5.33–5.34 and 7.40–7.50) . These gaps and misidentifications mean the model’s proof outline does not rigorously recover the paper’s quantitative result, even though it points in a broadly similar two-source-equidistribution direction.