Banach density of generated languages: Dichotomies in topology and dimension

The formalism of language generation in the limit studies generative models by requiring an algorithm, given strings from a hidden true language, to eventually generate new valid strings. A core issue is the tension between validity and breadth. Prior work quantified breadth via asymptotic density, where the priority is generating strings early in a natural countable ordering. Here, we study density when the strings are embedded in $d$ dimensions, a ubiquitous structure in current generative models. Our goal is for the generated strings to be dense throughout the embedding. This requires a different measure, the Banach density, which captures whether a set contains large sparse regions. Using Banach density uncovers a rich structure based on dimension and the topology of the language collection. We prove that in dimension one, when the underlying topological space has finite Cantor-Bendixson rank, an algorithm can always generate a subset of the true language with an optimal lower Banach density of 1/2. However, for collections with infinite Cantor-Bendixson rank, there are cases where no algorithm can achieve any positive lower Banach density; the generated set must contain arbitrarily large, sparse regions. This reveals a topological contrast unseen with asymptotic density, where 1/2 is always achievable. We also extend our results to a family of measures interpolating between Banach and asymptotic density. Finally, in dimension $d \geq 2$, our positive result for Banach density encounters a Ramsey-theoretic obstacle regarding two-colored point sets. Overcoming this requires a nondegeneracy condition: the embedding of the true language must be sufficiently represented throughout the full $d$-dimensional space.