A note on the distinct distances problem over finite fields
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909840688807936 |
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| author | Brukhim, Nataly Bruner, Ariel Raz, Orit E. |
| author_facet | Brukhim, Nataly Bruner, Ariel Raz, Orit E. |
| contents | We study a finite-field analogue of the Erdős distinct distances problem under the Hamming metric. For a set \(S\subseteq \mathbb{F}_q^n\) let $Δ(S)$ denote the set of Hamming distances determined by \(S\). We prove the lower bound \[ |Δ(S)| \;\ge\; \frac{\log |S|}{2\log(2nq)}, \] and show this bound is tight when \(|S|=O(\text{poly}(n))\), where the constant of proportionality depends only on $q$. We then also study the problem of finding a large \emph{rainbow set}, that is, a subset \(S\subseteq \mathbb{F}_q^n\) for which all \(\binom{|S|}{2}\) pairwise Hamming distances spanned by $S$ are distinct. In contrast to the Euclidean setting, we show that a set with many distinct distances does not imply the existence of a large rainbow set, by giving an explicit construction.
Nevertheless, we establish the existence of large rainbow sets, and prove that every large set in \(\mathbb{F}_q^n\) necessarily contains a non-trivial rainbow subset. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10869 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the distinct distances problem over finite fields Brukhim, Nataly Bruner, Ariel Raz, Orit E. Combinatorics Discrete Mathematics We study a finite-field analogue of the Erdős distinct distances problem under the Hamming metric. For a set \(S\subseteq \mathbb{F}_q^n\) let $Δ(S)$ denote the set of Hamming distances determined by \(S\). We prove the lower bound \[ |Δ(S)| \;\ge\; \frac{\log |S|}{2\log(2nq)}, \] and show this bound is tight when \(|S|=O(\text{poly}(n))\), where the constant of proportionality depends only on $q$. We then also study the problem of finding a large \emph{rainbow set}, that is, a subset \(S\subseteq \mathbb{F}_q^n\) for which all \(\binom{|S|}{2}\) pairwise Hamming distances spanned by $S$ are distinct. In contrast to the Euclidean setting, we show that a set with many distinct distances does not imply the existence of a large rainbow set, by giving an explicit construction. Nevertheless, we establish the existence of large rainbow sets, and prove that every large set in \(\mathbb{F}_q^n\) necessarily contains a non-trivial rainbow subset. |
| title | A note on the distinct distances problem over finite fields |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2510.10869 |