Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the exponents are $\delta$-separated.

Define $E_n$ to be the collection of all exponential polynomial of order $n$. i.e.,

$$ E_n:= \{ u : u(t) = \sum_{k=1}^n c_k e^{i \lambda_k t}, c_k \in \mathbb C, \lambda_k \in \mathbb R, and |\lambda_i - \lambda_j| > \delta\\ when\\ i \neq j \}. $$

Of course $u$ is an almost periodic function i.e., given $\epsilon >0$ there exists $T_\epsilon$ such that every interval of length $T_\epsilon$ contains an almost period of $u$. i.e., $\forall x \in \mathbb R,$ $\exists \tau \in (x,x+T_\epsilon)$ such that

$$ \sup_{t\in \mathbb R} |u(t)-u(t+\tau)| < \epsilon$$

Is it possible to find a bound on the $T_\epsilon$ for such an exponential polynomial, and may I even dare hope to find a bound which will work for the entire class maybe ? If that is too much to ask then what is a good question to ask ?

Has some one studied related questions or variations of it ? I would be glad to know.

Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the exponents are $\delta$-separated.

Define $E_n$ to be the collection of all exponential polynomial of order $n$. i.e.,

$$ E_n:= \{ u : u(t) = \sum_{k=1}^n c_k e^{i \lambda_k t}, c_k \in \mathbb C, \lambda_k \in \mathbb R, and |\lambda_i - \lambda_j| > \delta\\ when\\ i \neq j \}. $$

Of course $u$ is an almost periodic function i.e., given $\epsilon >0$ there exists $T_\epsilon$ such that every interval of length $T_\epsilon$ contains an almost period of $u$. i.e., $\forall x \in \mathbb R,$ $\exists \tau \in (x,x+T_\epsilon)$ such that

$$ \sup_{t\in \mathbb R} |u(t)-u(t+\tau)| < \epsilon$$

Is it possible to find a bound on the $T_\epsilon$ for such an exponential polynomial, I added the separation condition hoping that it would lead to an affirmative answer.

I have this suspicion that the almost period may have to do something with the number theoretical properties of the set of exponents. After all if all the exponents are in a lattice then the function is periodic.

and may I even dare hope to find a bound which will work for the entire class? Some nice subclass maybe ? 

If that is too much to ask then what is a good question to ask ?

Has some one studied related questions or variations of it ? I would be glad to know.