A sharpened form of the inequality for the concentration function

By means of the additive number theory the following sharpened form of Kesten's theorem for the concentration function is obtained.
Let $X_1,\dots,X_n$ be independent random variables,
$$ S_n=X_1+\dots+X_n,\ Q(X,\lambda)=\sup_x\mathbf P(x\le X\le x+\lambda). $$
Let $\lambda_j$, $1\le j\le n$, be any positive numbers such that $\lambda_j\ge 2\lambda$. Then
$$ Q(S_n,\lambda)\ll4\lambda\biggl[\sum_{j=1}^n\lambda_j^2(1-Q(X_j,\lambda_j))Q^{-2}(X_j,\lambda)\biggr]^{-1/2}. $$