Here is an issue I stumbled upon when trying to solve a problem. Keep in my that I reduced the more general problem to this one. For that reason, the statement could be false (but hopefully isn't).
Given a function $f\in L^p(\mathbb{R})$ (where $\mathbb{R}$ is equipped with the usual Lebesgue measure), is it true that for almost all $x\in\mathbb{R}$, the sequence defined by $$a_n=f(x-n)$$ is in $\ell^p$? In other words, does $$\sum_{n\in\mathbb{Z}} |a_n|^p=\sum_{n\in\mathbb{Z}} |f(x-n)|^p$$ converge for almost all $x\in\mathbb{R}$?
I have a strong feeling that the statement is true. I tried using the Dirac delta distribution by rewriting $$f(x-n)=\int_{-\infty}^{\infty} f(t)\delta(x-n-t)\;\mathrm{d}t.$$ Then, invoking the right theorems for permuting integrals and sums and for bounding the result, one could approach the problem of convergence by trying to show that $$\int_{-\infty}^\infty |f(t)|^p \Big(\sum_{n\in\mathbb{Z}}|\delta(x-n-t)|^p \Big) \;\mathrm{d}t$$ is convergent. I am not very familiar with distribution and looking a the sum of $\delta$'s above makes me uncomfortable.
I also thought about using the Poisson summation formula. This would make the translation $x\mapsto x-n$ into a multiplicative factor which seems easier to handle. I still have no clue about one could carry out this idea further without looping back to the previous idea about the Dirac $\delta$'s.
If you know of any well known identities which could give me a helping hand, that would be very appreciated.
Fix an $m\in\mathbb{Z}$, then \begin{align*} \int_{-\infty}^{\infty}|f(x)|^{p}dx&=\sum_{n\in\mathbb{Z}}\int_{n+m}^{n+m+1}|f(x)|^{p}dx\\ &=\sum_{n\in\mathbb{Z}}\int_{m}^{m+1}|f(x+n)|^{p}dx\\ &=\int_{m}^{m+1}\sum_{n\in\mathbb{Z}}|f(x+n)|^{p}dx, \end{align*} so \begin{align*} \sum_{n\in\mathbb{Z}}|f(x+n)|^{p}<\infty \end{align*} for a.e. $x\in[m,m+1]$. Now varying $m\in\mathbb{Z}$ to obtain a.e. result in $\mathbb{R}$.