2026/07/21 by Natalia Mora Cuellar, Ignacio Rojas Aravena, Alexia Yavicoli
#math.NT #math.DS
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 (Gn)n, a multiplicative Følner sequence (Fn)n, and any (α,β)∈[0,1]2, we construct a single set A⊆\N such that \dens(Gn)n(A)=α and \md(Fn)n(A)=β. 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.