A graph-theoretic proof of Cobham's Dichotomy for automatic sequences
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929348528832512 |
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| author | Wessel, Mieke |
| author_facet | Wessel, Mieke |
| contents | We give a new graph-theoretic proof of Cobham's Theorem which says that the support of an automatic sequence is either sparse or grows at least like $N^α$ for some $α> 0$. The proof uses the notions of tied vertices and cycle arboressences. With the ideas of the proof we can also give a new interpretation of the rank of a sparse sequence as the height of its cycle arboressence. In the non-sparse case we are able to determine the supremum of possible $α$, which turns out to be the logarithm of an integer root of a Perron number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11385 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A graph-theoretic proof of Cobham's Dichotomy for automatic sequences Wessel, Mieke Combinatorics Number Theory We give a new graph-theoretic proof of Cobham's Theorem which says that the support of an automatic sequence is either sparse or grows at least like $N^α$ for some $α> 0$. The proof uses the notions of tied vertices and cycle arboressences. With the ideas of the proof we can also give a new interpretation of the rank of a sparse sequence as the height of its cycle arboressence. In the non-sparse case we are able to determine the supremum of possible $α$, which turns out to be the logarithm of an integer root of a Perron number. |
| title | A graph-theoretic proof of Cobham's Dichotomy for automatic sequences |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2405.11385 |