An integral weight realization theorem for subset currents on free groups

We prove that if $N\ge 2$ and $\alpha: F_N\to \pi_1(\Gamma)$ is a marking on $F_N$, then for any integer $r\ge 2$ and any $F_N$-invariant collection of non-negative integral "weights" associated to all subtrees $K$ of $\widetilde \Gamma$ of radius $\le r$ satisfying some natural "switch" conditions, there exists a finite cyclically reduced folded $\Gamma$-graph $\Delta$ realizing these weights as numbers of "occurrences" of $K$ in $\Delta$. As an application, we give a new, more direct and explicit, proof of one of the main results of our paper with Nagnibeda \cite{KN3} stating that for any $N\ge 2$ the set $\gcnr$ of all rational subset currents is dense in the space $\gcn$ of subset currents on $F_N$. We also answer one of the questions (Problem 10.11) posed in \cite{KN3}. Thus we prove that if a nonzero $\mu\in \gcn$ has all weights with respect to some marking being integers, then $\mu$ is the sum of finitely many "counting" currents corresponding to nontrivial finitely generated subgroups of $F_N$.