Bound for the maximal probability in the Littlewood-Offord problem

The paper deals with studying a connection of the Littlewood--Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. It is shown that the values at zero of the concentration functions of weighted sums of i.i.d. random variables may be estimated by the values at zero of the concentration functions of symmetric infinitely divisible distributions with the L\'evy spectral measures which are multiples of the sum of delta-measures at $\pm$weights involved in constructing the weighted sums.