Polynomial porosity decay in the degree

Determine whether, for fixed dimension n≥3, fixed closed Riemannian target N, BLD constant L, linear local contractibility constant λ, and contractibility scale r_*, the porosity bound for branch values of degree-m branched covers can be improved, at an m-independent positive scale threshold, to c m^{-α} for constants c,α>0 independent of m.

Background

Theorem 6.1 provides a porosity constant of the form δ_m=1/2(a/(m+1))^{m-1} at a scale threshold depending only on the fixed geometric data and the target. Theorem 1.8 establishes the sharp order m^{-1/2} for sphere-to-sphere covers in dimension two, while product constructions show that slower decay cannot generally be excluded in higher-dimensional manifold classes.

The unresolved issue is whether the exponential-in-m estimate supplied by the general recursive argument can be replaced, for fixed n≥3 and the stated data, by a polynomial lower bound c m^{-α} uniformly over all degrees and covers.