AI 中文总结
该研究解决了阿贝尔群有限子集的和差不等式中指数最优性的公开问题，构造集族证明第一个不等式的指数1/2最优，相关工作借助AI智能体Hyra完成。
AI 中文摘要
对于阿贝尔群的有限非空子集A，令σ(A)=|A+A|/|A|，δ(A)=|A−A|/|A|。经典和差不等式指出σ(A)^(1/2)≤δ(A)≤σ(A)^2，已知第二个不等式中的指数2是最优的，而第一个不等式中的指数1/2能否改进则一直是公开问题。我们解决了该问题，通过构造有限集族A_K⊂Z，使得logσ(A_K)/logδ(A_K)趋近于2，因此第一个不等式中的指数1/2也是最优的。该构造及其证明借助了基于开放权重Hy3模型的AI研究智能体Hyra的协助。
英文摘要
For a finite nonempty subset $A$ of an abelian group, let $σ(A)=|A+A|/|A|$ and $δ(A)=|A-A|/|A|$. The classical sum-difference inequalities state that $$σ(A)^{1/2}\leqδ(A)\leqσ(A)^2.$$ The exponent $2$ in the second inequality is known to be optimal, whereas it has remained open whether the exponent $1/2$ in the first inequality can be improved. We settle this question by constructing an explicit family of finite sets $A_K\subset\mathbb{Z}$ such that $$\frac{\logσ(A_K)}{\logδ(A_K)}\longrightarrow 2,$$ hence the exponent $1/2$ in the first inequality is also optimal. The construction and its proof were developed with the assistance of Hyra, an AI research agent based on the open-weights Hy3 model.
