Geodesic
Estimates for the concentration functions in the Littlewood--Offord problem
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 20, Tome 420 (2013), pp. 50-69

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Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\sum_{k=1}^na_kX_k$ with respect to the arithmetic structure of coefficients $a_k$. Such concentration results recently became important in connection with investigations about singular values of random matrices. In this paper we formulate and prove some refinements of a result of Vershynin (2011). 
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     author = {Yu. S. Eliseeva and F. G\"otze and A. Yu. Zaitsev},
     title = {Estimates for the concentration functions in the {Littlewood--Offord} problem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {50--69},
     publisher = {mathdoc},
     volume = {420},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a2/}
}
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Yu. S. Eliseeva; F. Götze; A. Yu. Zaitsev. Estimates for the concentration functions in the Littlewood--Offord problem. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 20, Tome 420 (2013), pp. 50-69. http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a2/