发表机构
Università degli Studi di Brescia（布雷西亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文通过结合两个熵不等式，将恢复对乘积的上界从(2.2499)^n改进至(2.23949)^n，证明了新的更紧的指数界。
AI 中文摘要
我们证明了每个由n元集合的子集组成的非空族恢复对(\mathcal{A},\mathcal{B})满足|\mathcal{A}||\mathcal{B}|\le(223949/100000)^n=(2.23949)^n。Mond、Souza和Versteegen的一个结果，结合Fang和Huang对消去对的尖锐(9/4)^n界，给出了恢复对的(2.2499)^n界。我们将其改进为(2.23949)^n。证明结合了两个熵不等式。一个直接由恢复性质导出，另一个将Fang-Huang界应用于由原始对通过Mond、Souza和Versteegen的构造得到的族。然后通过凸组合将这两个不等式结合起来。
英文摘要
We prove that every recovering pair $(\mathcal{A},\mathcal{B})$ of nonempty families of subsets of an $n$-element set satisfies $$|\mathcal{A}||\mathcal{B}|\le\left(\frac{223949}{100000}\right)^n=(2.23949)^n.$$ A result of Mond, Souza and Versteegen, combined with the sharp $(9/4)^n$ bound of Fang and Huang for cancellative pairs, gives $(2.2499)^n$ for recovering pairs. We improve this to $(2.23949)^n$. The proof combines two entropy inequalities. One is derived directly from the recovering property, while the other applies the Fang-Huang bound to families obtained from the original pair by a construction of Mond, Souza and Versteegen. The two inequalities are then combined by a convex combination.
CommentsComputational supplement, verifiers and outputs are available at https://doi.org/10.5281/zenodo.23093260
