On the application of Arak's inequalities for concentration functions to the Littlewood-Offord problem

We discuss the results of a joint paper of Eliseeva, Götze and Zaitsev. At the beginning of 1980's, Arak (1980, 1981) has published new bounds for the concentration functions of sums of independent random variables. These bounds were formulated in terms of the arithmetic structure of supports of distributions of summands. Using these results, he has obtained the final solution of an old problem posed by Kolmogorov (1956). In this talk, we apply Arak's results to the Littlewood-Offord problem which was intensively investigated in the last years. We compare the consequences of Arak's results with recent results of Nguyen, Tao and Vu.