Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices

We consider $N\times N$ non-Hermitian random matrices of the form $X+A$, where $A$ is a general deterministic matrix and $\sqrt{N}X$ consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, that is, that the local density of eigenvalues is bounded by $N^{1+o\left(1\right)}$ and (ii) that the expected condition number of any bulk eigenvalue is bounded by $N^{1+o\left(1\right)}$; both results are optimal up to the factor $N^{o\left(1\right)}$. The latter result complements the very recent matching lower bound obtained by Cipolloni et al. and improves the $N$-dependence of the upper bounds by Banks et al. and Jain et al. Our main ingredient, a near-optimal lower tail estimate for the small singular values of $X+A-z$, is of independent interest.