论文标题
关于最大的无额整数集的注释
A note on the largest sum-free sets of integers
论文作者
论文摘要
给定$ a $ a $ n $正整数，一个添加剂组合的一个旧问题要求$ a $是否包含一个不含额的大小的子集，至少$ n/3+ω（n）$，对于某些增加的无绑定函数$ω$。在文献中通常通过考虑另一种猜想来攻击这个问题，该猜想认为是$ n \ to \ infty $，$ \ max_ {x \ in \ mathbb {r}/\ mathbb {z}}} \ sum_ \ sum_ \ sum_ {n \ 1} _ {（1/3,2/3）} -1/3）（nx）\ to \ infty $。如果是真的，这个猜想也意味着每$ k \ geq1 $的$（2K，4K）$ - 无汇总设置发生类似现象。在本说明中，我们直接证明了后者的结果。我们证明的新成分是对主机设置$ a $的结构分析，这可能具有独立的兴趣。
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.
