Arithmetic regularity as an alternative to transference
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910282026057728 |
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| author | Chow, Sam Prendiville, Sean Vazquez, Santiago |
| author_facet | Chow, Sam Prendiville, Sean Vazquez, Santiago |
| contents | Since Green (2005), the Fourier-analytic transference principle has dominated the landscape of combinatorial theorems relative to sparse arithmetic sets. We demonstrate a different approach using arithmetic regularity. This is more versatile and has the potential to succeed when no obvious `dense model' is forthcoming. Moreover, we contend that, just as the traditional circle method disassembles an arithmetic problem into real and $p$-adic parts which can be solved individually, the arithmetic regularity method generalises this to yield an additional `combinatorial' factor. This framework leads directly to a correct lower bound on the number of configurations in a dense set. We illustrate this using a system comprising a linear equation together with a higher-degree equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02312 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Arithmetic regularity as an alternative to transference Chow, Sam Prendiville, Sean Vazquez, Santiago Number Theory 11B30 (primary), 11D45, 11D72, 11L15 (secondary) Since Green (2005), the Fourier-analytic transference principle has dominated the landscape of combinatorial theorems relative to sparse arithmetic sets. We demonstrate a different approach using arithmetic regularity. This is more versatile and has the potential to succeed when no obvious `dense model' is forthcoming. Moreover, we contend that, just as the traditional circle method disassembles an arithmetic problem into real and $p$-adic parts which can be solved individually, the arithmetic regularity method generalises this to yield an additional `combinatorial' factor. This framework leads directly to a correct lower bound on the number of configurations in a dense set. We illustrate this using a system comprising a linear equation together with a higher-degree equation. |
| title | Arithmetic regularity as an alternative to transference |
| topic | Number Theory 11B30 (primary), 11D45, 11D72, 11L15 (secondary) |
| url | https://arxiv.org/abs/2606.02312 |