On the rate of convergence in Kolmogorov's uniform limit theorem. I

Theorem. {\it For any probability distribution function $F$ on $R$ and for any natural number $n$ there exists an infinitely divisible distribution function $B$ such that
$$ \sup_x|F^{n*}(x)-B(x)|\le C_n^{-2/3} $$
} Here $F^{n*}$ is the $n$-fold convolution of $F$ with itself and $C$ is an absolute constant. The paper contains the first part of the proof.