Local semicircle law under fourth moment condition

We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that $\max_{jk} {\mathbb E} |X_{jk}|^{4+\delta} < \infty, \delta > 0$, it was proved in [G\"otze, Naumov and Tikhomirov, Bernoulli, 2018] that with high probability the typical distance between the Stieltjes transforms $m_n(z), z = u + i v$, of the empirical spectral distribution (ESD) and the Stieltjes transforms $m_{sc}(z)$ of the semicircle law is of order $(nv)^{-1} \log n$. The aim of this paper is to remove $\delta>0$ and show that this result still holds if we assume that $\max_{jk} {\mathbb E} |X_{jk}|^{4} < \infty$. We also discuss applications to the rate of convergence of the ESD to the semicircle law in the Kolmogorov distance, rates of localization of the eigenvalues around the classical positions and rates of delocalization of eigenvectors.