SMALL DIAMETERS AND GENERATORS FOR ARITHMETIC LATTICES IN SL2(R) AND CERTAIN RAMANUJAN GRAPHS

The paper proves that Γ0(Q) is generated by elements of Frobenius norm Oε(Q^{1+ε}) and, for co-compact arithmetic lattices Γ, that almost every conjugate σ^{-1}Γσ is generated by elements of Frobenius norm Oε(V_Γ^{2+ε}), with uniformity under additional hypotheses; these are stated explicitly as Theorems 1–3 in the introduction and proved via a spectral averaging method that bounds a covering radius and then uses side-pairing generators of a fundamental domain to turn displacement bounds into Frobenius-norm bounds . The candidate solution reaches the same claims by: (i) equating Frobenius norm with hyperbolic displacement (consistent with u(z,w)=¼ tr(gg^t)−½ in the paper), (ii) using side-pairings to generate Γ, and (iii) bounding a covering radius by a pre-trace/L^2 argument to obtain displacements ≪ log V_Γ, hence Frobenius norms ≪ V_Γ^{2+ε}. This mirrors the paper’s logic: spectral averaging to show that balls of radius T≈(2+o(1))log V_Γ cover almost surely, then side-pairings generate, and finally norm–distance converts to Frobenius bounds . Minor differences are technical tools (pre-trace kernel vs. sphere-averaging operator) and the model’s slightly stronger intermediate heuristic for Γ0(Q) (“all entries ≪ Q”), whereas the paper proves Oε(Q^{1+ε}) via Ford/Farey analysis of isometric circles; but the final generator bounds coincide .