Almost periodic functions have been defined and studied by the Danish mathematician Harald Bohr. Bohr's motivations were number theory and Dirichlet series. A trigonometric polynomial $S(x)$ is a function of the real variable $x$ defined by $ S(x)=\sum _1^N c_k\exp (2\pi i \lambda _k x)$ were the frequencies $\lambda _k$ are arbitrary real numbers and the coefficients $c_k$ are real or complex numbers. An almost periodic function in the sense given by Bohr is a uniform limit on $\mathbb R$ of a sequence $S_m$ of trigonometric polynomials. This definition leads to the computation or estimation of $\sup _{x\in {\mathbb R}}|S(x)|=\|S\|_{\infty }$ which is often a hard problem. Let us define $T(\epsilon )>0$ as the lower bound of the set of $|x|$ such that $|S(x)|\geq (1-\epsilon )\|S\|_{\infty }.$ The estimation of $T(\epsilon )$ as $\epsilon \to 0$ depends on the arithmetical property of the set of frequencies $\lambda _k.$ This booklet is devoted to these issues which are illustrated on three examples.