Almost periodic sequence

Remarks

Example: a periodic sequence with period $p$ is almost periodic. Every multiple of $p$ is an $\varepsilon$-almost period for every $\varepsilon > 0$, and every set of $p$ consecutive integers contains a multiple of $p$, so $L = p$ works.

Non-example: let $x_n = 0$ for $n < 0$ and $x_n = 1$ for $n \ge 0$. For $p > 0$ take $n = -1$, and for $p < 0$ take $n = 0$: in both cases $\abs{x_{n+p} - x_n} = 1$. So for $\varepsilon = 1/2$ the only almost period is $0$, and no $L$ works.

Equivalent forms: a bounded sequence is almost periodic if and only if the set of its translates $(x_{n+p})_{n \in \Z}$, $p \in \Z$, is relatively compact for the supremum norm. It is also equivalent that the sequence is a uniform limit on $\Z$ of trigonometric polynomials $\sum_{j=1}^{m} c_j e^{i \lambda_j n}$ with $c_j \in \C$ and $\lambda_j \in \R$. An almost periodic sequence is bounded.