  In this short note, we extend the celebrated results of Tao and Vu, and
Krishnapur on the universality of empirical spectral distributions to a wide
class of inhomogeneous complex random matrices, by showing that a technical and
hard-to-verify Fourier domination assumption may be replaced simply by a
natural uniform anti-concentration assumption.
  Along the way, we show that inhomogeneous complex random matrices, whose
expected squared Hilbert-Schmidt norm is quadratic in the dimension, and whose
entries (after symmetrization) are uniformly anti-concentrated at $0$ and
infinity, typically have smallest singular value $\Omega(n^{-1/2})$. The rate
$n^{-1/2}$ is sharp, and closes a gap in the literature.
  Our proofs closely follow recent works of Livshyts, and Livshyts, Tikhomirov,
and Vershynin on inhomogeneous real random matrices. The new ingredient is a
couple of anti-concentration inequalities for sums of independent, but not
necessarily identically distributed, complex random variables, which may also
be useful in other contexts.
