On the concentration function
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 371-375

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The characteristic function has the property that, if it is large at two points, it is large at the sum of these points too. Using this fact, a number of properties of characteristic functions is proved. In particular, a new proof of a theorem due to Rogozin is given which estimates the concentration function in the case of lattice summands.
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L. P. Postnikova; A. A. Yudin. On the concentration function. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 371-375. http://geodesic.mathdoc.fr/item/TVP_1977_22_2_a11/