On lower bounds of exponential frames in l1 norm Let $\{t_k\}_{k=-\infty}^\infty$ be a sequence of real numbers.
I'm interested in finding the largest number A such that
\begin{equation*}
\int_{-\Omega}^\Omega|\sum_{k=-\infty}^{+\infty}c_ke^{2\pi i t_kf}|df\ge A \sum_{k=-\infty}^{+\infty}|c_k|
\end{equation*}
holds for every sequence $\{c_k\}_{k=-\infty}^\infty$. Clearly, $A$ depends on the sequence $t_k$. In particular I'm interested in the case where the sequence $t_k$ are separated. That is, 
\begin{equation*}
\min_{k\neq\ell}|t_k-t_\ell|\ge \delta.
\end{equation*}
So I would like to calculate $A$ as a function of $\delta$.
 A: Here's something in the vein of a counterexample, at least to what you're conjecturing. Let $r > 1$, and consider the sequence $t_{k} = r k / \Omega$, which has $\min_{k \neq \ell} | t_{k} - t_{l} | = r / \Omega > 1 / \Omega$. By changing variable in the integral, the inequality becomes
$$
\frac{\Omega}{r}
\int_{-r}^{r}
\biggl|
\sum_{k} c_{k} e^{2 \pi i k x}
\biggr|
\, dx
\geq
A \sum_{k} |c_{k}|.
$$
This inequality in turn implies
$$
\frac{2 \Omega \lceil r \rceil}{r}
\int_{0}^{1}
\biggl|
\sum_{k} c_{k} e^{2 \pi i k x}
\biggr|
\, dx
\geq
A \sum_{k} |c_{k}|.
$$
I'm not familiar with the intricacies of convergence of Fourier series, but this latter inequality is essentially of the form
$$
\| \hat{g} \|_{1}
\leq C \| g \|_{1}
$$
for functions $g$ on the unit circle $\mathbb{T}$, and no such inequality holds.
A: As Julien (in the comments) and Jason (in the previous answer) point out, the inequality you want is impossible, which is easily seen by considering the case of a Dirichlet kernel. Also I am perplexed by comment about a result of Arne Beurling. Clearly, the left hand side of your proposed inequality is non-negative, so the result is trivial with $A=0$. 
However, there is an much weaker inequality of the form you want. The inequality states (for $\Omega >1/2$) that:
$$\int_{-\Omega}^\Omega|\sum_{k=1}^{\infty}c_ke^{2\pi i t_k x}|dx \ge A \sum_{k=1}^{\infty}\frac{|c_k|}{k} $$
where $t_{k+1}-t_{k} \geq 1$ and $A$. Note that you can rescale to consider the case of smaller $\delta$ (after adjusting $\Omega$ appropriately).
This was proven by Fedja Nazarov in On a proof of the Littlewood conjecture by McGehee, Pigno and Smith. Algebra i Analiz 7 (1995), no.2, pp. 106-120.
As the title suggests, the proof is a generalization of the McGehee-Pigno-Smith proof of Littlewood's conjecture on exponential sum which is precisely the case with $\Omega =1/2$ and the $t_i$'s are taken to be distinct integers. 
One can see that the left side of Nazarov's inequality is of the order $\log(n)$ in the case of the Dirichlet kernel of size $n$. Thus, apart from the implicit constant, this inequality is best possible.
