  Three dimensional topological insulators are bulk insulators with
$\mathbf{Z}_2$ topological electronic order that gives rise to conducting
light-like surface states. These surface electrons are exceptionally resistant
to localization by non-magnetic disorder, and have been adopted as the basis
for a wide range of proposals to achieve new quasiparticle species and device
functionality. Recent studies have yielded a surprise by showing that in spite
of resisting localization, topological insulator surface electrons can be
reshaped by defects into distinctive resonance states. Here we use numerical
simulations and scanning tunneling microscopy data to show that these resonance
states have significance well beyond the localized regime usually associated
with impurity bands. At native densities in the model Bi$_2$X$_3$ (X=Bi, Te)
compounds, defect resonance states are predicted to generate a new quantum
basis for an emergent electron gas that supports diffusive electrical
transport.
