Shortening the distance between Edgeworth and Berry-Esseen in the classical case

We obtain near optimal Berry-Esseen bounds for standardized sums of independent identically distributed random variables. This is achieved by distinguishing the lattice and the non-lattice cases, as one-term Edgeworth expansions do. The main tool is an easy inequality involving the usual second modulus of continuity, in substitution of Esseen's smoothing inequality. An illustrative example concerning the exponential distribution is also considered.