Hello,


I have come across the function


$f(t) = \sum\_{j=1}^n c\_j e^{2 \pi i a\_j t}$


with $c\_j \in \mathbb{C}$, $c\_j\neq 0$ and $a\_j\in\mathbb{R}$, $a\_j \neq 0$ for $j=1,...,n$, and the $a\_j$ distinct. I want to show that $f(t)$ is periodic with least period equal to $1/\gcd a\_j$ if the $a\_j$ have a common divisor, and not periodic otherwise.


Is anyone aware of a good reference for this question?



Least common period of a finite sum of exponentials
For functions of this form, define an inner product by $\langle f,g\rangle=\lim\_{T\to\infty}\frac1T\int\_0^T f(t)\bar g(t)\,dt$. With this inner product, the set of functions $e^{2\pi i at}$ form an uncountable orthogonal set.


If $f(t)$ is periodic with period $s$, then $f(t)=f(t+s)=\sum\_{j=1}^n (e^{2\pi i a\_js}c\_j)e^{2\pi i a\_jt}$. Since the inner product of $f$ with itself is $\sum |c\_j|^2$, we deduce that $e^{2\pi i a\_js}=1$ for each $j$, proving the claim.


