Attainable forms of Assouad spectra

The PDF proves a full classification of attainable Assouad spectra (Theorem A) with the equivalence (a)⇔(b)⇔(c), via: (a)⇒(b) by a standard two-scale covering inequality (Proposition 3.1), (c)⇒(b) by verifying the cap family Cd satisfies the same inequality (Proposition 3.6), (closure under suprema, Proposition 3.7), and (b)⇒(a) by constructing homogeneous Moran sets (Proposition 3.3, Theorem 3.5) and a canonical sup-of-caps representation (Theorem 3.11). The candidate solution follows the same overall structure: two-scale covering for (a)⇒(b), a sup-of-caps functional description for (b)⇒(c), and a Moran-type construction and sup closure for (c)⇒(a). Minor gaps remain in the candidate’s Step 2 (the cap selection/touching argument is asserted rather than established, whereas the paper provides a precise c(κ,λ,y) parametrization and lemmas ensuring global domination), but the approach and main claims match the paper’s results.