Is the space of reachable particle configurations dense?
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918107680866304 |
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| author | Pach, Janos Tardos, Gabor |
| author_facet | Pach, Janos Tardos, Gabor |
| contents | Let $p_0,\ldots,p_n$ be a finite sequence of points in an Euclidean space $\R^d$. Suppose that there is a (pointlike) particle sitting at each point $p_i$. In a ``legal'' move, any one of them can jump over another, landing on the other side, at exactly the same distance. Under what circumstances can we guarantee that for any $\varepsilon>0$ and any other sequence of points $q_0,\ldots, q_n\in\R^d$, there is a finite sequence of legal moves that takes the particle at $p_i$ to the $\varepsilon$-neighborhood of $q_i$, simultaneously for every $i$?
We prove that this is possible if and only if the additive group generated by the vectors $p_1-p_0,\ldots,p_n-p_0$ is dense in $\R^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22471 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Is the space of reachable particle configurations dense? Pach, Janos Tardos, Gabor Combinatorics 52C07 (primary) 15A04, 05B35 (secondary) G.2.3; G.2.1 Let $p_0,\ldots,p_n$ be a finite sequence of points in an Euclidean space $\R^d$. Suppose that there is a (pointlike) particle sitting at each point $p_i$. In a ``legal'' move, any one of them can jump over another, landing on the other side, at exactly the same distance. Under what circumstances can we guarantee that for any $\varepsilon>0$ and any other sequence of points $q_0,\ldots, q_n\in\R^d$, there is a finite sequence of legal moves that takes the particle at $p_i$ to the $\varepsilon$-neighborhood of $q_i$, simultaneously for every $i$? We prove that this is possible if and only if the additive group generated by the vectors $p_1-p_0,\ldots,p_n-p_0$ is dense in $\R^d$. |
| title | Is the space of reachable particle configurations dense? |
| topic | Combinatorics 52C07 (primary) 15A04, 05B35 (secondary) G.2.3; G.2.1 |
| url | https://arxiv.org/abs/2507.22471 |