Fractional part sums and lattice points

The objective of this article are sums S(M)= Sigma(n) psi(Mf(n/M)) where psi denotes essentially the fractional part minus 1/2, f is a C-4-function with f " nonvanishing, and summation is extended over an interval of order M. For S(M) an R-estimate and a mean-square bound is obtained. Applications to problems concerning the number of lattice points in large planar domains are discussed.