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_kt_\ell\ge \delta. \end{equation*} So I would like to calculate $A$ as a function of $\delta$.

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. 


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 nonnegative, 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. 106120. As the title suggests, the proof is a generalization of the McGeheePignoSmith 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. 

