Geodesic
A~sharpened form of the inequality for the concentration function
Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 2, pp. 376-379

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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 $$ Q(S_n,\lambda)\ll4\lambda\biggl[\sum_{j=1}^n\lambda_j^2(1-Q(X_j,\lambda_j))Q^{-2}(X_j,\lambda)\biggr]^{-1/2}. $$ 
@article{TVP_1978_23_2_a9,
     author = {L. P. Postnikova and A. A. Yudin},
     title = {A~sharpened form of the inequality for the concentration function},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {376--379},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1978_23_2_a9/}
}
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L. P. Postnikova; A. A. Yudin. A~sharpened form of the inequality for the concentration function. Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 2, pp. 376-379. http://geodesic.mathdoc.fr/item/TVP_1978_23_2_a9/