🤖 AI Summary
This study investigates the natural density of subsets in free groups, with a focus on rational subsets and finitely generated subgroups. By introducing the notion of relative density from formal language theory and integrating it with irreducible shifts of finite type and the language of freely reduced words, the authors develop a linguistic framework to analyze density behavior. Their main contributions include the first language-theoretic proof that automorphism orbits have natural density zero, a complete characterization of rational subsets possessing positive density, and the demonstration that a subgroup has positive density if and only if it has finite index. Furthermore, in non-convergent cases, they establish weak convergence properties and prove convergence of the average of upper and lower densities.
📝 Abstract
In this paper, we study the density of subsets of nonabelian free groups using relative densities of languages. We start by proving some basic properties about the density of a language $L_1$ relative to another language $L_2$ containing $L_1$. We then focus on the case where $L_2$ is the language of freely reduced words over an alphabet and prove an analogue of the Infinite Monkey Theorem for this language. This result, obtained as a corollary of a broader theorem on irreducible subshifts of finite type, allows for a language-theoretic characterization of rational subsets with positive density. As a consequence, we obtain a language-theoretic proof that the automorphic orbit of an element of a nonabelian free group has natural density zero, which generalizes a result by Burillo and Ventura concerning the density of primitive elements of free groups. We then describe rational subsets of free groups of positive density and explore in depth the case of finitely generated subgroups. We prove that if the rational subset is a subgroup, then it has positive density if and only if it has finite index and characterize those for which there is convergence. In cases where convergence fails due to parity constraints, we show that the density of the subset always exhibits weak convergence and that the average of its supremum and infimum densities converges to the expected value.