The probabilities of large deriations on Borel sets

Accuracy of the approximation of the probability $P_n(A_n)=\mathbf P(\frac1{\sqrt n}(X_1+\dots+X_n)\in A_n)$ by $\Phi(A_n)$ is studied for Borel sets $A_n$, $\Phi(A_n)\to0$. The necessary and sufficient conditions are obtained for $P_n(A_n)=\Phi(A_n)(1+O(\ae(\sqrt n)))$ uniformly in all sequences $\{A_n\}$ such that $\Phi(A_n)\geqslant\Phi(x:|x|>\bar\Lambda(\sqrt n))$. Here $\ae(z)\downarrow0$, $\bar\Lambda(z)\uparrow\infty$ are functions satisfying some conditions.