  Due to the effort of a number of authors, the value c_u of the absolute
constant factor in the uniform Berry--Esseen (BE) bound for sums of independent
random variables has been gradually reduced to 0.4748 in the iid case and
0.5600 in the general case; both these values were recently obtained by
Shevtsova. On the other hand, Esseen had shown that c_u cannot be less than
0.4097. Thus, the gap factor between the best known upper and lower bounds on
(the least possible value of) c_u is now rather close to 1.
  The situation is quite different for the absolute constant factor c_{nu} in
the corresponding nonuniform BE bound. Namely, the best correctly established
upper bound on c_{nu} in the iid case is about 25 times the corresponding best
known lower bound, and this gap factor is greater than 30 in the general case.
In the present paper, improvements to the prevailing method (going back to S.
Nagaev) of obtaining nonuniform BE bounds are suggested. Moreover, a new method
is presented, of a rather purely Fourier kind, based on a family of smoothing
inequalities, which work better in the tail zones. As an illustration, a quick
proof of Nagaev's nonuniform BE bound is given. Some further refinements in the
application of the method are shown as well.
