通过联合亏缺缩放引理证明16维的迪特猜想
Dittert's conjecture in dimension 16 via a joint-deficit scaling lemma
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中文总结 AI 辅助
研究迪特猜想在n = 16时的情况，核心方法是利用联合亏缺估计、平斯克型子集和界限、克诺普 - 辛克霍恩边界下界及黄的正支持定理，主要贡献是证明了n = 16时该猜想，结合其他结果确立了\(n\geq16\)时迪特猜想。
中文摘要 AI 辅助
迪特猜想断言，在元素和为n的非负n×n矩阵中，函数\(\phi(A)=\prod_{i = 1}^n r_i+\prod_{j = 1}^n c_j-\text{per}(A)\)在均匀矩阵\(J_n/n\)处唯一最大化。本文证明了n = 16时的该猜想。关键在于对于接近最大化者，行和与列和乘积的亏缺满足单个联合约束而非两个独立界限。结合联合亏缺估计与平斯克型子集和界限，得到对双超随机矩阵更精确的标量扩张。然后利用永久性的克诺普 - 辛克霍恩边界下界排除有零元素的最大化者，通过黄的正支持定理确定唯一最大化者。与庞对于\(n\geq17\)的结果一起，确立了\(n\geq16\)时的迪特猜想。
英文摘要
Dittert's conjecture asserts that, among nonnegative $n\times n$ matrices whose entries sum to $n$, the functional $ϕ(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\operatorname{per}(A)$ is uniquely maximized by the uniform matrix $J_n/n$. This paper proves the conjecture for $n=16$. The key observation is that, for a near-maximizer, the deficits of the row-sum and column-sum products satisfy a single joint constraint rather than two independent bounds. Combining this joint-deficit estimate with a Pinsker-type subset-sum bound yields a sharper scalar dilation to a doubly superstochastic matrix. The Knopp-Sinkhorn boundary lower bound for permanents then excludes maximizers with a zero entry, and Hwang's positive-support theorem identifies the unique maximizer. Together with Pang's result for $n\ge 17$ (arXiv:2606.01531), this establishes Dittert's conjecture for every $n\ge 16$.
