  We develop a theory for the compressibility of the surface states of 3D
topological insulators and propose that surface probes of the compressibility
via scanning single electron transistor microscopy will be a straightforward
way to access the topological states without interference from the bulk states.
We describe the single-particle nature of the surface states taking into
account an accurate Hamiltonian for the bands and then include the contribution
from electron--electron interactions and discuss the implications of the
ultra-violet cutoff, including the universality of the exchange contribution
when expressed in dimensionless units. We also compare the theory with
experimentally obtained d{\mu}/dn as extracted from angle-resolved
photoemission spectroscopy measurements. Finally, we point out that
interaction-driven renormalization of the Fermi velocity may be discernible via
this technique.
