Burgess-Type Bounds for Character Sums over $\mathbb{F}_{p^n}$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911597115473920 |
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| author | Chattopadhyay, Aishik |
| author_facet | Chattopadhyay, Aishik |
| contents | We establish Burgess-type bounds for short multiplicative character sums over finite fields $\mathbb{F}_{p^n}$ under a purely volumetric condition. We show that for a box $B \subset \mathbb{F}_{p^n}$, nontrivial cancellation occurs whenever $|B| \ge p^{n(1/4+\varepsilon)}$, without imposing lower bounds on the individual side lengths.
This removes the coordinate-wise restrictions present in earlier results and extends work of Gabdullin for $n=2,3$ to arbitrary dimension. The proof combines methods from the geometry of numbers with multiplicative energy estimates and bounds for character sums due to Katz. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22167 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Burgess-Type Bounds for Character Sums over $\mathbb{F}_{p^n}$ Chattopadhyay, Aishik Number Theory 11L26, 11L40 We establish Burgess-type bounds for short multiplicative character sums over finite fields $\mathbb{F}_{p^n}$ under a purely volumetric condition. We show that for a box $B \subset \mathbb{F}_{p^n}$, nontrivial cancellation occurs whenever $|B| \ge p^{n(1/4+\varepsilon)}$, without imposing lower bounds on the individual side lengths. This removes the coordinate-wise restrictions present in earlier results and extends work of Gabdullin for $n=2,3$ to arbitrary dimension. The proof combines methods from the geometry of numbers with multiplicative energy estimates and bounds for character sums due to Katz. |
| title | Burgess-Type Bounds for Character Sums over $\mathbb{F}_{p^n}$ |
| topic | Number Theory 11L26, 11L40 |
| url | https://arxiv.org/abs/2602.22167 |