Edit distance in substitution systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916955298988032 |
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| author | Best, Andrew Peres, Yuval |
| author_facet | Best, Andrew Peres, Yuval |
| contents | Let $σ$ be a primitive substitution on an alphabet $\mathcal{A}$, and let $\mathcal{W}_n$ be the set of words of length $n$ determined by $σ$ (i.e., $w \in \mathcal{W}_n$ if $w$ is a subword of $σ^k(a)$ for some $a \in \mathcal{A}$ and $k \geq 1$). It is known that the corresponding substitution dynamical system is loosely Kronecker (also known as zero-entropy loosely Bernoulli), so the diameter of $\mathcal{W}_n$ in the edit distance is $o(n)$. We improve this upper bound to $O(n/\sqrt{\log n})$. The main challenge is handling the case where $σ$ is non-uniform; a better bound is available for the uniform case. Finally, we show that for the Thue--Morse substitution, the diameter of $\mathcal{W}_n$ is at least $\sqrt {n/6} - 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_00440 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Edit distance in substitution systems Best, Andrew Peres, Yuval Combinatorics Dynamical Systems 37B10 (Primary) 68R15 (Secondary) Let $σ$ be a primitive substitution on an alphabet $\mathcal{A}$, and let $\mathcal{W}_n$ be the set of words of length $n$ determined by $σ$ (i.e., $w \in \mathcal{W}_n$ if $w$ is a subword of $σ^k(a)$ for some $a \in \mathcal{A}$ and $k \geq 1$). It is known that the corresponding substitution dynamical system is loosely Kronecker (also known as zero-entropy loosely Bernoulli), so the diameter of $\mathcal{W}_n$ in the edit distance is $o(n)$. We improve this upper bound to $O(n/\sqrt{\log n})$. The main challenge is handling the case where $σ$ is non-uniform; a better bound is available for the uniform case. Finally, we show that for the Thue--Morse substitution, the diameter of $\mathcal{W}_n$ is at least $\sqrt {n/6} - 1$. |
| title | Edit distance in substitution systems |
| topic | Combinatorics Dynamical Systems 37B10 (Primary) 68R15 (Secondary) |
| url | https://arxiv.org/abs/2501.00440 |