  Let $N_n$ be an $n\times n$ complex random matrix, each of whose entries is
an independent copy of a centered complex random variable $z$ with finite
non-zero variance $\sigma^{2}$. The strong circular law, proved by Tao and Vu,
states that almost surely, as $n\to \infty$, the empirical spectral
distribution of $N_n/(\sigma\sqrt{n})$ converges to the uniform distribution on
the unit disc in $\mathbb{C}$.
  A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from
additive combinatorics, is controlling the lower tail of the least singular
value of the random matrix $xI - N_{n}/(\sigma\sqrt{n})$ (where $x\in
\mathbb{C}$ is fixed) with failure probability that is inverse polynomial. In
this paper, using a simple and novel approach (in particular, not using tools
from additive combinatorics or any net arguments), we show that for any fixed
matrix $M$ with operator norm at most $n^{0.51}$ and for all $\eta \geq 0$,
$$\Pr\left(s_n(M+N_n) \leq \eta \right) \lesssim n^{C}\eta + \exp(-n^{c}),$$
where $s_n(M+N_n)$ is the least singular value of $M+N_n$ and $C,c$ are
absolute constants. Our result is optimal up to the constants $C,c$ and the
inverse exponential-type error rate improves upon the inverse polynomial error
rate due to Tao and Vu.
  During the course of our proof, we extend the solution of the counting
problem in inverse Littlewood-Offord theory, recently isolated by the author
along with Ferber, Luh, and Samotij, from Rademacher variables to general
complex random variables. This significantly improves on estimates for this
problem obtained using the optimal inverse Littlewood-Offord theorem of Nguyen
and Vu, and may be of independent interest.
