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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.14229 |
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| _version_ | 1866916461565444096 |
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| author | Mandelshtam, Andrei |
| author_facet | Mandelshtam, Andrei |
| contents | Ulam words are binary words defined recursively as follows: the length-$1$ Ulam words are $0$ and $1$, and a binary word of length $n$ is Ulam if and only if it is expressible uniquely as a concatenation of two shorter, distinct Ulam words. We discover, fully describe, and prove a surprisingly rich structure already in the set of Ulam words containing exactly two $1$'s. In particular, this leads to a complete description of such words and a logarithmic-time algorithm to determine whether a binary word with two $1$'s is Ulam. Along the way, we uncover delicate parity and biperiodicity properties, as well as sharp bounds on the number of $0$'s outside the two $1$'s. We also show that sets of Ulam words indexed by the number $y$ of $0$'s between the two $1$'s have intricate tensor-based hierarchical structures determined by the arithmetic properties of $y$. This allows us to construct an infinite family of self-similar Ulam-word-based fractals indexed by the set of $2$-adic integers, containing the outward Sierpinski gasket as a special case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14229 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On fractal patterns in Ulam words Mandelshtam, Andrei Combinatorics Ulam words are binary words defined recursively as follows: the length-$1$ Ulam words are $0$ and $1$, and a binary word of length $n$ is Ulam if and only if it is expressible uniquely as a concatenation of two shorter, distinct Ulam words. We discover, fully describe, and prove a surprisingly rich structure already in the set of Ulam words containing exactly two $1$'s. In particular, this leads to a complete description of such words and a logarithmic-time algorithm to determine whether a binary word with two $1$'s is Ulam. Along the way, we uncover delicate parity and biperiodicity properties, as well as sharp bounds on the number of $0$'s outside the two $1$'s. We also show that sets of Ulam words indexed by the number $y$ of $0$'s between the two $1$'s have intricate tensor-based hierarchical structures determined by the arithmetic properties of $y$. This allows us to construct an infinite family of self-similar Ulam-word-based fractals indexed by the set of $2$-adic integers, containing the outward Sierpinski gasket as a special case. |
| title | On fractal patterns in Ulam words |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2211.14229 |