A large part of twentieth century analytic number theory was devoted to finding good estimates for these sums, a trend started by basic work of Hermann Weyl in diophantine approximation.
The best that is reasonable to hope for is an estimate of the form which signifies, up to the implied constant in the big O notation, that the sum resembles a random walk in two dimensions.
Even such a minor-seeming result in the right direction has to be referred all the way back to the structure of the initial sequence xn, to show a degree of randomness.
A variant of 'Weyl differencing' investigated by Weyl involving a generating exponential sum
was previously studied by Weyl himself, he developed a method to express the sum as the value
If the sum is of the form where ƒ is a smooth function, we could use the Euler–Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S(x), then for large values of a you could use "stationary phase" method to calculate the integral and give an approximate evaluation of the sum.
The Weil conjectures had major applications to complete sums with domain restricted by polynomial conditions (i.e., along an algebraic variety over a finite field).
A basic advance was Weyl's inequality for such sums, for polynomial f. There is a general theory of exponent pairs, which formulates estimates.
This is the ideal degree of cancellation one could hope for without any a priori knowledge of the structure of the sum, since it matches the scaling of a random walk.
The sum of exponentials is a useful model in pharmacokinetics (chemical kinetics in general) for describing the concentration of a substance over time.