  We obtain Dini conditions with "exponent 2" that guarantee that an
asymptotically conformal quasisphere is rectifiable. In particular, we show
that for any e>0 integrability of
  (esssup_{1-t < |x| < 1+t} K_f(x)-1)^{2-e} dt/t implies that the image of the
unit sphere under a global quasiconformal homeomorphism f is rectifiable. We
also establish estimates for the weak quasisymmetry constant of a global
K-quasiconformal map in neighborhoods with maximal dilatation close to 1.
