The elements of the closure of $\{ \sum\_{j=1}^n a\_j e^{i\nu\_j x}: a\_j\in \mathbb{C}, \nu\_j\in \mathbb{R} \}$ in the supremum-norm are called almost periodic functions. An almost periodic function $f$ is called quasi-periodic iff the frequency module


$$ \mathcal{M}(f)=\left\langle \left\{ \nu\in \mathbb{R}: \lim\_{t\rightarrow \infty} \frac{1}{t} \int\_0^t f(x)e^{-i\nu x}dx\neq 0 \right\} \right\rangle\_{\mathbb{Z}} $$


is finitely generated as $\mathbb{Z}$-module. By definition we can approximate $f$ in supremum-norm by functions of the form $\sum\_{j=1}^n a\_j e^{i\nu\_j x}$. I'd like to know whether one can preserve the frequencies.


**Question**: Is it always possible to approximate $f$ in the supremum-norm by functions $\sum\_{j=1}^n a\_j e^{i\nu\_j x}$ with $\nu\_j\in \mathcal{M}(f)$?



Approximation of quasi-periodic function by trigonometric polynomials
Yes. This is Satz XIV, p. 160 of Harald Bohr, [Zur Theorie der Fastperiodischen Funktionen: II, Acta Math. **46** (1925) 101-214](https://projecteuclid.org/euclid.acta/1485887596). (Page 162 attributes the quasi-periodic case to Bohl.)


