Representation of quasi-periodic functions and Hausdorff-Young inequalities for Besicovitch almost periodic functions

For a class of ${\mathbb{R}}^d$-ations and ${\mathbb{Z}}^d$-actions on the $n$-dimensional torus ${\mathbb{T}}^n$, we characterize their unique ergodicity and establish a theorem of Weyl type. This result allows us to establish an isomorphism between the Banach algebra of quasi-periodic functions with spectrum in a given $\mathbb{Z}$-module and the Banach algebra of periodic functions on a torus. This, in return, allows us to give a very simple proof of Hausdorff-Young inequalities for Besicovitch almost periodic functions. The regularity of the parent function of a quasi-periodic function is also studied.