EFFECTIVE EQUIDISTRIBUTION OF TRANSLATES OF TORI IN ARITHMETIC HOMOGENEOUS SPACES AND APPLICATIONS

The uploaded paper proves the exact asymptotic N_{3,p}(T)=μ_{A\G}(B_T)+O(T^{3−κ}) for cubic p splitting over R (Theorem 1.1) and derives it via a new, technically involved effective equidistribution of translates of tori (Theorem 1.3) and a careful reduction in Section 8, not by a generic spectral-gap/wavefront argument . The model’s outline asserts the same conclusion but relies on tools (wavefront property and generic quantitative mixing) that apply to affine symmetric spaces, not to the A\G setting for n=3; the paper explicitly contrasts this and introduces new machinery to overcome the lack of wavefront in this case . Hence the paper is correct, while the model’s justification is not.