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

$$ \mathcal{M}(f)=\left\langle \left\{ \nu\in \mathbb{R}: \lim_{t\rightarrow \infty} \int_0^t f(x)e^{-i\nu x}dx\neq 0 \right\} \right\rangle_{\mathbb{Z}} $$

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 $\nu_j\in \mathcal{M}(f)$?

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 $\nu_j\in \mathcal{M}(f)$?