Sum-difference problem III（AlphaEvolve）

问题

Let $C$ be the supremum of all constants such that there exist arbitrarily large finite sets of integers $A,B$ with $|A+B| \lesssim |A|$ and $|A-B| \gtrsim |A|^{C}$. Establish upper and lower bounds for $C$ that are as strong as possible.

问题陈述引自 AlphaEvolve 问题库的第 44 题（Google DeepMind，CC BY 4.0），保留英文原文。

AI 做了什么

AlphaEvolve 曾改进了最好的已知界，这一纪录后来又被超越；问题本身仍未解决。问题库中的状态：曾经的新结果，后来被超越。此后纪录可能已有变化，下面问题库的注释列出了它所知的最新界。

问题库中的注释（英文原文）

Improved constructions were subsequently obtained in "Sums and differences of sets (improvement over AlphaEvolve)" by Robert Gerbicz and in "Sums and differences of sets: a further improvement over AlphaEvolve" by Fan Zheng. by

如何验证

这些结果都是明确的构造或程序；问题库中的 notebook 附有用来打分的验证代码。

局限

改进一个界是进展，而不是解答：问题的确切答案仍然未知。作者也报告说，由专家设置和引导搜索时效果更好。

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