Strong Orthogonality between the Mobius Function and Nonlinear Exponential Functions in Short Intervals

Let mu(n) be the Mobius function, e(z)= exp(2 pi iz), x real and 2 <= y <= x. This paper proves two sequences (mu(n)) and (e(n(k)alpha)) are strongly orthogonal in short intervals. That is, if k >= 3 being fixed and y >= x(1-1/4+epsilon), then for any A > 0, we have Sigma(x< n <= X+Y) mu(n)e(n(k)alpha) << Y(log Y)(-A) uniformly for alpha is an element of R.