On the rate of decay of the concentration function of the sum of independent random variables (Q2574054)

Let \(X_1, X_2, \ldots\) be i.i.d. integer-valued random variables. The paper discusses two problems concerning the rate of decay of the concentration function of \(X_1+\cdots+X_n\). The results obtained are refinements of an assertion due to \textit{J. M. Deshouillers, G. A. Freiman} and \textit{A. A. Yudin} [in: Structure theory of set addition. Astérisque 258, 425--436 (1999; Zbl 0944.60028)]. In particular, it is shown that in that assertion certain arithmetic property of the underlying distribution cannot be dispensed with.