Product ${\displaystyle {\mathbb{Z}}^d}$-actions on a Lebesgue space and their applications

We define a class of ${\displaystyle {\mathbb{Z}}^d}$-actions, d ≥ 2, called product ${\displaystyle {\mathbb{Z}}^d}$-actions. For every such action we find a connection between its spectrum and the spectra of automorphisms generating this action. We prove that for any subset A of the positive integers such that 1 ∈ A there exists a weakly mixing ${\displaystyle {\mathbb{Z}}^d}$-action, d≥2, having A as the set of essential values of its multiplicity function. We also apply this class to construct an ergodic ${\displaystyle {\mathbb{Z}}^d}$-action with Lebesgue component of multiplicity ${\displaystyle 2^{d}k}$, where k is an arbitrary positive integer.