Additive and multiplicative densities, prime valuations and symbolic models

We study additive and multiplicative densities of subsets of $\mathbb{N}$ along prescribed F{\o}lner sequences. We prove that additive upper density one implies multiplicative density one along a suitable multiplicative F{\o}lner sequence. We also prove independence in the following sense: given an additive F{\o}lner sequence $(G_n{)}_n$, a multiplicative F{\o}lner sequence $(F_n{)}_n$, and any $(\alpha ,\beta )\in [0,1{]}^2$, we construct a single set $A\subseteq \mathbb{N}$ such that ${\text{dens}}_{(G_n{)}_n}(A)=\alpha$ and ${\text{md}}_{(F_n{)}_n}(A)=\beta$. Prime-valuation coordinates yield exact density formulas and random models for one local condition and, under summability assumptions, countably many; in the finite-coordinate case they also give exact higher-order correlations. Finally, we realize these multiplicative correlations as correlations in symbolic dynamical systems and obtain criteria for ergodicity and mixing.