The Erdős-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields
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| Format: | Preprint |
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2025
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| author | Kang, Hunseok Koh, Doowon Rakhmonov, Firdavs |
| author_facet | Kang, Hunseok Koh, Doowon Rakhmonov, Firdavs |
| contents | In this paper, we study the cardinality of the distance set $Δ(A, B)$ determined by two subsets $A$ and $B$ of the $d$-dimensional vector space over a finite field $\mathbb{F}_q$. Assuming that $A$ or $B$ lies in a $k$-coordinate plane up to translations and rotations, we prove that if $|A||B| > 2q^d$, then $|Δ(A, B)| > q/2$, where $|Δ(A, B)|$ denotes the number of distinct distances between elements of $A$ and $B$. In particular, we show that our result recovers the sharp $(d+1)/2$ threshold for the Erdős-Falconer distance problem in odd dimensions, where distances are determined by a single set. As an application, we also obtain an improved result on the Box distance problem posed by Borges, Iosevich, and Ou, in the case where $2$ is a square in $\mathbb{F}_q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_07251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Erdős-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields Kang, Hunseok Koh, Doowon Rakhmonov, Firdavs Combinatorics Number Theory primary: 42B10, 52C10, secondary: 11T23 In this paper, we study the cardinality of the distance set $Δ(A, B)$ determined by two subsets $A$ and $B$ of the $d$-dimensional vector space over a finite field $\mathbb{F}_q$. Assuming that $A$ or $B$ lies in a $k$-coordinate plane up to translations and rotations, we prove that if $|A||B| > 2q^d$, then $|Δ(A, B)| > q/2$, where $|Δ(A, B)|$ denotes the number of distinct distances between elements of $A$ and $B$. In particular, we show that our result recovers the sharp $(d+1)/2$ threshold for the Erdős-Falconer distance problem in odd dimensions, where distances are determined by a single set. As an application, we also obtain an improved result on the Box distance problem posed by Borges, Iosevich, and Ou, in the case where $2$ is a square in $\mathbb{F}_q$. |
| title | The Erdős-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields |
| topic | Combinatorics Number Theory primary: 42B10, 52C10, secondary: 11T23 |
| url | https://arxiv.org/abs/2506.07251 |