  We study the planar Coulomb gas in the regime where the inverse temperature
$\beta_n$ grows at least logarithmically with respect to the number of
particles $n$ (freezing regime, $\beta_n\gtrsim \log n$). We show that, almost
surely for large $n$, the discrepancy between the number of particles in any
microscopic region and their expected value (given with adequate precision by
the equilibrium measure) is, up to log factors, of the order of the perimeter
of the observation window. The estimates are valid throughout the whole droplet
(the region where the particles accumulate), and are particularly interesting
near the boundary, while in the bulk they offer technical improvements over
known results.
  Our work builds on recent results on equidistribution at low temperatures and
improves on them by providing refined spectral asymptotics for certain Toeplitz
operators on the range of the erfc-kernel (sometimes called Faddeeva or plasma
dispersion kernel).
