Two dimensional arithmetic progressions avoiding squares
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914555467137024 |
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| author | Dietmann, Rainer Elsholtz, Christian |
| author_facet | Dietmann, Rainer Elsholtz, Christian |
| contents | We show that any proper symmetric two dimensional arithmetic progression contained in the interval $[-T,T]$ which avoids non-zero perfect squares has at most $O_\varepsilon(T^{20/27+\varepsilon})$ elements. This improves on a result of Croot, Lyall and Rice. We also discuss lower bounds for this problem and their connections to bounds for the least quadratic non-residue modulo a prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11104 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Two dimensional arithmetic progressions avoiding squares Dietmann, Rainer Elsholtz, Christian Number Theory Primary: 11B25, Secondary: 11D09, 11P70 We show that any proper symmetric two dimensional arithmetic progression contained in the interval $[-T,T]$ which avoids non-zero perfect squares has at most $O_\varepsilon(T^{20/27+\varepsilon})$ elements. This improves on a result of Croot, Lyall and Rice. We also discuss lower bounds for this problem and their connections to bounds for the least quadratic non-residue modulo a prime. |
| title | Two dimensional arithmetic progressions avoiding squares |
| topic | Number Theory Primary: 11B25, Secondary: 11D09, 11P70 |
| url | https://arxiv.org/abs/2605.11104 |