Parabolic distance in $\mathbb F_q^2$: a sharp exponent and new results
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| Format: | Preprint |
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2026
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| _version_ | 1866917356692832256 |
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| author | Van Anh, Dao Nguyen Senger, Steven Tran, Dung The Vinh, Le Anh |
| author_facet | Van Anh, Dao Nguyen Senger, Steven Tran, Dung The Vinh, Le Anh |
| contents | We study the parabolic variant of the Erd\H os--Falconer distance problem in finite fields. That is, if $q$ is odd, we seek size thresholds beyond which any subset $E\subset \mathbb F_q^2$ will determine many distinct parabolic distances. This problem has a rich history because the parabolic distance functional shares many properties with the standard distance functional, but exhibits many distinct behaviors. Here we begin with rather standard Fourier analytic arguments, but diverge into additive combinatorics to handle the central obstructions. We provide a suite of positive results and corresponding sharpness examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21292 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Parabolic distance in $\mathbb F_q^2$: a sharp exponent and new results Van Anh, Dao Nguyen Senger, Steven Tran, Dung The Vinh, Le Anh Combinatorics 52C10, 11T23, 42B10, 05B25 We study the parabolic variant of the Erd\H os--Falconer distance problem in finite fields. That is, if $q$ is odd, we seek size thresholds beyond which any subset $E\subset \mathbb F_q^2$ will determine many distinct parabolic distances. This problem has a rich history because the parabolic distance functional shares many properties with the standard distance functional, but exhibits many distinct behaviors. Here we begin with rather standard Fourier analytic arguments, but diverge into additive combinatorics to handle the central obstructions. We provide a suite of positive results and corresponding sharpness examples. |
| title | Parabolic distance in $\mathbb F_q^2$: a sharp exponent and new results |
| topic | Combinatorics 52C10, 11T23, 42B10, 05B25 |
| url | https://arxiv.org/abs/2603.21292 |