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.

翻译：极限语言生成的形式化研究通过要求算法从隐藏的真实语言中给定字符串后最终生成新的有效字符串来研究生成模型。核心问题在于有效性与广度之间的张力。先前工作通过渐近密度量化广度，其优先考虑在自然可数排序中尽早生成字符串。本文研究字符串嵌入$d$维空间（当前生成模型中的普遍结构）时的密度问题，目标是使生成字符串在整个嵌入空间中稠密。这需要采用不同的度量——Banach密度，该密度可刻画集合是否包含大的稀疏区域。利用Banach密度揭示了基于维度与语言集合拓扑的丰富结构。我们证明：在一维空间中，当底层拓扑空间具有有限Cantor-Bendixson秩时，算法总能生成真实语言的子集，并达到最优下半Banach密度1/2。然而，对于具有无限Cantor-Bendixson秩的集合，存在算法无法实现任何正下半Banach密度的情况——生成集必然包含任意大的稀疏区域。这揭示了渐近密度中始终可达到1/2时未见的拓扑对比。我们还将结果推广至插值于Banach密度与渐近密度之间的一族测度。最后，在维度$d \geq 2$时，关于Banach密度的正结果遭遇涉及双色点集的Ramsey理论障碍。克服该障碍需要非退化条件：真实语言的嵌入必须充分覆盖整个$d$维空间。