Exponential sum

Remarks

Each term has absolute value $1$, so $\abs{S} \le \#A$; this is the trivial bound. Example: for $f(n) = n\alpha$ with $\alpha \in \R \setminus \Z$ and $A = \set{1, \dots, N}$, the sum is geometric and $$\abs{S} = \abs{\frac{e(N\alpha) - 1}{e(\alpha) - 1}} \le \frac{1}{\abs{\sin \pi \alpha}},$$ a bound independent of $N$. Non-example: $\sum_{n=1}^{N} 2^n$ is a sum of values of an exponential function, but it is not an exponential sum in this sense. Its terms have absolute value larger than $1$, so they are not of the form $e(f(n))$ with $f$ real. When $f$ is a polynomial, $S$ is called a Weyl sum. Sums with weights, $\sum_{n \in A} a_n e(f(n))$ with complex $a_n$, are also called exponential sums.