On the calculation of constants in the Arak inequality for the concentration functions of convolution of probability distributions

The aim of present work is evaluation of the absolute constants in the Arak inequalities for the concentration functions of convolutions of probability distributions. This result will subsequently allow us to calculate the constant in the inequality for the uniform distance between $ n $ and \break $(n + 1)$-fold convolutions of one-dimensional symmetric probability distributions with a characteristic function separated from $-1$, as well as a number of other estimates, in particular, the accuracy of the approximation of samples of rare events by the Poisson point process.