On the completeness induced by densities on natural numbers

Abstract: Let $\nu: \mathcal{P}(\mathbb{N}) \to \mathbb{R}$ be an \textquotedblleft upper density\textquotedblright\, on the natural numbers $\mathbb{N}$ (for instance, $\nu$ can be the upper asymptotic density or the upper Banach density). Then a natural pseudometric $d_\nu$ is induced on $\mathcal{P}(\mathbb{N})$, namely, $$ \forall A,B\subseteq \mathbb{N}, \quad d_\nu(A,B):=\nu(A\bigtriangleup B) $$ We provide necessary and sufficient conditions for the completeness of $d_\nu$. Then we identify in which cases the latter ones are verified.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)