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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.18383 |
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| _version_ | 1866908395629445120 |
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| author | Becher, Verónica Carton, Olivier Figueira, Santiago |
| author_facet | Becher, Verónica Carton, Olivier Figueira, Santiago |
| contents | In 1976, Rauzy studied two complexity functions, $\underlineβ$ and $\overlineβ$, for infinite sequences over a finite alphabet. The function $\underlineβ$ achieves its maximum precisely for Borel normal sequences, while $\overlineβ$ reaches its minimum for sequences that, when added to any Borel normal sequence, result in another Borel normal sequence. We establish a connection between Rauzy's complexity functions, $\underlineβ$ and $\overlineβ$, and the notions of non-aligned block entropy, $\underline{h}$ and $\overline{h}$, by providing sharp upper and lower bounds for $\underline{h}$ in terms of $\underlineβ$, and sharp upper and lower bounds for $\overline{h}$ in terms of $\overlineβ$. We adopt a probabilistic approach by considering an infinite sequence of random variables over a finite alphabet. The proof relies on a new characterization of non-aligned block entropies, $\overline{h}$ and $\underline{h}$, in terms of Shannon's conditional entropy. The bounds imply that sequences with $\overline{h} = 0$ coincide with those for which $\overlineβ = 0$. We also show that the non-aligned block entropies, $\underline{h}$ and $\overline{h}$, are essentially subadditive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rauzy dimension and finite-state dimension Becher, Verónica Carton, Olivier Figueira, Santiago Information Theory Formal Languages and Automata Theory In 1976, Rauzy studied two complexity functions, $\underlineβ$ and $\overlineβ$, for infinite sequences over a finite alphabet. The function $\underlineβ$ achieves its maximum precisely for Borel normal sequences, while $\overlineβ$ reaches its minimum for sequences that, when added to any Borel normal sequence, result in another Borel normal sequence. We establish a connection between Rauzy's complexity functions, $\underlineβ$ and $\overlineβ$, and the notions of non-aligned block entropy, $\underline{h}$ and $\overline{h}$, by providing sharp upper and lower bounds for $\underline{h}$ in terms of $\underlineβ$, and sharp upper and lower bounds for $\overline{h}$ in terms of $\overlineβ$. We adopt a probabilistic approach by considering an infinite sequence of random variables over a finite alphabet. The proof relies on a new characterization of non-aligned block entropies, $\overline{h}$ and $\underline{h}$, in terms of Shannon's conditional entropy. The bounds imply that sequences with $\overline{h} = 0$ coincide with those for which $\overlineβ = 0$. We also show that the non-aligned block entropies, $\underline{h}$ and $\overline{h}$, are essentially subadditive. |
| title | Rauzy dimension and finite-state dimension |
| topic | Information Theory Formal Languages and Automata Theory |
| url | https://arxiv.org/abs/2406.18383 |