New applications of Arak's inequalities to the Littlewood-Offord problem
Let X-1, . . . , X-n be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums Sigma(n)(k=1) X(k)a(k) with respect to the arithmetic structure of coefficients a(k) in the context of the Littlewood-Offord problem. In recent papers of Eliseeva, Gotze and Zaitsev, we discussed the relations between the inverse principles stated by Nguyen, Tao and Vu and similar principles formulated by Arak in his papers from the 1980's. In this paper, we will derive some more general and more precise consequences of Arak's inequalities providing new results in the Littlewood-Offord problem.
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Springer International Publishing AG