On triple correlations of Fourier coefficients of cusp forms. II

The triple correlation Sigma(n is an element of(X,2X]) alpha(n)beta(n-h)gamma(n+h) for arbitrary coefficients alpha,beta,gamma is estimated on average over h in some short intervals. By introducing a device of short intervals, we are able to reduce this problem to uniform oscillations of one of the three coefficients, say gamma, against additive characters of R/Z over short intervals. The argument is simple, but refines previous arguments in certain cases. More precise estimates are also obtained by taking gamma to be Fourier coefficients of cusp forms and Mobius functions, which substantially improve previous results.