On Fourier Asymptotics and Effective Equidistribution

The paper proves that if the Fourier ℓ1-dimension of µ exceeds 39/64 = 1/2 + 7/64, then µy equidistributes effectively with a rate y^η for any 0 < η < dimℓ1 µ − 39/64. This threshold is explicitly tied to the best-known bound toward Ramanujan (7/64), and the argument hinges on a uniform bound sup_m |φ̂y(m)| ≪ |y|^{1/2−θ} for θ > 7/64, together with truncation in the m-variable and control by Sobolev norms . The candidate solution reaches the same threshold and rate by a more direct spectral/Whittaker–Fourier analysis: inserting µ̂(m) against the K-Bessel expansions and summing with Hecke bounds |λ(m)| ≪ m^{θ+ε}, then controlling the spectral sum via Sobolev weights and Weyl’s law. The methods are closely related but organized differently (the paper proves a strong uniform bound for φ̂y(m) first; the model bounds the µ-average on each eigenfunction directly), and both yield µy(φ) − mX(φ) ≪ y^{α−1/2−θ+o(1)} under dimℓ1 µ = α > 1/2 + θ. Minor imprecision in the model’s discussion of Eisenstein constant terms does not affect correctness, since those terms already decay like y^{1/2} and are dominated by the cusp contribution .