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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2311.11162 |
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| _version_ | 1866916911509405696 |
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| author | Bell, Jason Gorman, Alexi Block |
| author_facet | Bell, Jason Gorman, Alexi Block |
| contents | This paper concerns the expansion of the real ordered additive group by a predicate for a subset of $[0,1]$ whose base-$r$ representations are recognized by a Büchi automaton. In the case that this predicate is closed, a dichotomy is established for when this expansion is interdefinable with the structure $(\mathbb{R},<,+,0,r^{-\mathbb{N}})$ for some $r \in \mathbb{N}_{>1}$. In the case that the closure of the predicate has Hausdorff dimension less than $1$, the dichotomy further characterizes these expansions of $(\mathbb{R},<,+,0,1)$ by when they have NIP and NTP$_2$, which is precisely when the closure of the predicate has Hausdorff dimension $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11162 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sparse regular subsets of the reals Bell, Jason Gorman, Alexi Block Logic Formal Languages and Automata Theory 03C64, 03D05, 28A80 This paper concerns the expansion of the real ordered additive group by a predicate for a subset of $[0,1]$ whose base-$r$ representations are recognized by a Büchi automaton. In the case that this predicate is closed, a dichotomy is established for when this expansion is interdefinable with the structure $(\mathbb{R},<,+,0,r^{-\mathbb{N}})$ for some $r \in \mathbb{N}_{>1}$. In the case that the closure of the predicate has Hausdorff dimension less than $1$, the dichotomy further characterizes these expansions of $(\mathbb{R},<,+,0,1)$ by when they have NIP and NTP$_2$, which is precisely when the closure of the predicate has Hausdorff dimension $0$. |
| title | Sparse regular subsets of the reals |
| topic | Logic Formal Languages and Automata Theory 03C64, 03D05, 28A80 |
| url | https://arxiv.org/abs/2311.11162 |