Additive and multiplicative densities, prime valuations and symbolic models

We study additive and multiplicative densities of subsets of $\N$ along prescribed Følner sequences.

We prove that additive upper density one implies multiplicative density one along a suitable multiplicative Følner sequence.

We also prove independence in the following sense: given an additive Følner sequence $(G_n)_n$, a multiplicative Følner sequence $(F_n)_n$, and any $(\alpha,\beta)\in[0,1]^2$, we construct a single set $A\subseteq\N$ such that $\dens_{(G_n)_n}(A)=\alpha$ and $\md_{(F_n)_n}(A)=\beta$.

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.