Consider the family of functions $$ \forall n \in \mathbb N,\quad \forall x \in \mathbb R,\quad e_n(x) = e^{in x}, $$ which is known to be a Riesz basis of $L^2(0,2 \pi)$. What happens when we look at this family in $L^2(0,L)$ for $ 0 < L < 2 \pi$? I am mostly interested in the $\omega$ independence in $L^2(0,L)$ with $\ell^2$ coefficients. Namely: is it true that for any sequence $(a_n)_{n=0}^\infty \in \ell^2( \mathbb N)$, if $$ \sum_{n=0}^\infty a_n e_n = 0 $$ in $L^2(0,L)$, then $a_n = 0$ for all $n \in \mathbb N$?
Note that for any $(a_n)_{n=0}^\infty \in \ell^2( \mathbb N)$ the series $\sum a_n e_n$ converges in $L^2(0,2 \pi)$ because there the $(e_n)$ are orthogonal. Therefore, $\sum a_n e_n$ converges in $L^2(0,L)$ and in the above question we do not have to assume the convergence.
An easy situation is when $\sum a_n e_n$ extends to a holomorphic function on $$ \Omega_\epsilon := \{ z \in \mathbb C : 0 < \Re z < 2 \pi,\quad - \epsilon < \Im z < \epsilon \} $$ for some $\epsilon > 0$. Then in view of the isolated zeros principle we get $\sum a_n e_n = 0$ on $(0,2\pi)$ hence $a_n = 0$ for all $n$.
This situation arises, for instance, when the series $\sum a_n e^{inz}$ converges normally on $\Omega_\epsilon$, that is when $$ \sum_{n=0}^\infty |a_n| e^{n\epsilon} < \infty. $$
In case $(a_n)_{n=0}^\infty$ is merely square summable it is not clear to me whether the $L^2_{loc}(\mathbb R)$ function $\sum_{n=0}^\infty a_n e_n$ has an analytic extension to $\mathbb C$ hence I am not convinced I can adapt the above arguments.
Observe that there already exists several closely related results in the litterature. The Kadec $1/4$ lemma gives a condition for a non-harmonic series to be a Riesz basis, the Duffin and Schaeffer theorem gives a condition for a non-harmonic series to be a frame. But these concepts of Riesz basis and frame are respectively stronger and not related to $\omega$-independence. This makes me suspect that what I ask is somehow harder than those results.
A quick example of a sequence $\{a_n\}$ which is not identically zero but satisfies $\sum_{n=0}^\infty a_n e_n = 0$ on all of $(0,L)$ is the Fourier series of $u(x-L)$, where $u$ is the unit step function. This shows that although the $e_n$ span all of $L^2(0,L)$, they are not a basis because they are not linearly independent.
This suggests two further questions. Does any strict subset of the $e_n$ span $L^2(0,L)$? How small can we make $E$ such that $\{e_n\}\setminus E$ is linearly independent? The answers are easy for $2\pi/L\in\mathbb Z$, but otherwise it's less clear.