Three Questions of Erdős-Nathanson on Asymptotic Bases of Order 2
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912941630029824 |
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| author | Larsen, Daniel |
| author_facet | Larsen, Daniel |
| contents | We study three natural properties that measure the robustness of asymptotic bases of order 2: having divergent representation function, being decomposable as a union of two bases, and containing a minimal basis. Erdős and Nathanson showed that sufficiently rapid growth of the representation function (specifically, $r_A(n) \ge C \log n$ for appropriate $C$) implies both decomposability and the existence of a minimal basis. We prove that for weaker growth rates, these three properties are independent. The construction proceeds via an inductive scheme on exponentially growing intervals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_03472 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Three Questions of Erdős-Nathanson on Asymptotic Bases of Order 2 Larsen, Daniel Number Theory 11B13 (Primary) We study three natural properties that measure the robustness of asymptotic bases of order 2: having divergent representation function, being decomposable as a union of two bases, and containing a minimal basis. Erdős and Nathanson showed that sufficiently rapid growth of the representation function (specifically, $r_A(n) \ge C \log n$ for appropriate $C$) implies both decomposability and the existence of a minimal basis. We prove that for weaker growth rates, these three properties are independent. The construction proceeds via an inductive scheme on exponentially growing intervals. |
| title | Three Questions of Erdős-Nathanson on Asymptotic Bases of Order 2 |
| topic | Number Theory 11B13 (Primary) |
| url | https://arxiv.org/abs/2603.03472 |