A note on the largest sum-free sets of integers
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866916132045193216 |
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| author | Jing, Yifan Wu, Shukun |
| author_facet | Jing, Yifan Wu, Shukun |
| contents | Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+ω(N)$ for some increasing unbounded function $ω$. The question is generally attacked in the literature by considering another conjecture, which asserts that as $N\to\infty$, $\max_{x\in\mathbb{R}/\mathbb{Z}}\sum_{n\in A}({\bf 1}_{(1/3,2/3)}-1/3)(nx)\to\infty$. This conjecture, if true, would also imply that a similar phenomenon occurs for $(2k,4k)$-sum-free sets for every $k\geq1$. In this note, we prove the latter result directly. The new ingredient of our proof is a structural analysis on the host set $A$, which might be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_09963 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A note on the largest sum-free sets of integers Jing, Yifan Wu, Shukun Combinatorics Classical Analysis and ODEs Number Theory 11B30, 11K70, 05D10 Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+ω(N)$ for some increasing unbounded function $ω$. The question is generally attacked in the literature by considering another conjecture, which asserts that as $N\to\infty$, $\max_{x\in\mathbb{R}/\mathbb{Z}}\sum_{n\in A}({\bf 1}_{(1/3,2/3)}-1/3)(nx)\to\infty$. This conjecture, if true, would also imply that a similar phenomenon occurs for $(2k,4k)$-sum-free sets for every $k\geq1$. In this note, we prove the latter result directly. The new ingredient of our proof is a structural analysis on the host set $A$, which might be of independent interest. |
| title | A note on the largest sum-free sets of integers |
| topic | Combinatorics Classical Analysis and ODEs Number Theory 11B30, 11K70, 05D10 |
| url | https://arxiv.org/abs/2011.09963 |