AI 中文总结
针对林德斯特伦唯一和问题，本文开发坐标投影方法，结合Ordentlich等人的必要条件松弛形式，首次将Lindström的上界3/2严格改进为约1.4884。
AI 中文摘要
令{0,1}^n表示所有分量为0或1的n维向量构成的集合。对于任意两个非空子集𝒳,𝒴⊆{0,1}^n，若每个对(x,y)∈𝒳×𝒴都可由算术和x+y唯一确定，则称(𝒳,𝒴)为n维唯一和对。设M(n)为所有n维唯一和对(𝒳,𝒴)中|𝒳||𝒴|的最大值。1969年，Lindström证明了1/2(1+log₂3) ≤ limₙ→∞(1/n)log₂M(n) ≤ 3/2。此后，下界(1+log₂3)/2陆续得到改进，而上界3/2始终未变。本文中，我们建立了一个数值约为1.4884的显式上界，据我们所知，这是首次对Lindström的上界3/2进行严格改进。为此，我们开发了一种坐标投影方法，可从一个唯一和对构造出低维唯一和系统；将该方法的结果与Ordentlich和Shayevitz建立的关于唯一和系统的必要条件的松弛形式相结合，我们得到了上述改进后的上界。
英文摘要
Let $\{0,1\}^{n}$ denote the set of all $n$-dimensional vectors whose components are either $0$ or $1$. For any two nonempty subsets $\mathcal{X},\mathcal{Y}\subseteq\{0,1\}^{n}$, the pair $(\mathcal X,\mathcal Y)$ is called an $n$-dimensional unique-sum pair if each pair $(\boldsymbol x,\boldsymbol y)\in\mathcal X\times\mathcal Y$ can be uniquely determined from the arithmetic sum $\boldsymbol x+\boldsymbol y$. Let $M(n)$ denote the maximum value of $|\mathcal X||\mathcal Y|$ over all $n$-dimensional unique-sum pairs $(\mathcal X,\mathcal Y)$. In 1969, Lindström proved that $$ \frac{1}{2}(1+\log_2 3) \leq \lim_{n\to\infty}\frac{1}{n}\log_2 M(n) \leq 3/2. $$ Since then, the lower bound $(1+\log_2 3)/2$ has been successively improved, whereas the upper bound $3/2$ has remained unchanged. In this paper, we establish an explicit upper bound whose numerical value is approximately $1.4884$. To the best of our knowledge, this is the first strict improvement over Lindström's upper bound $3/2$. Towards this end, we develop a coordinate projection approach that constructs a lower-dimensional unique-sum system from a unique-sum pair. By combining the results obtained from this approach with a relaxed form of a necessary condition established by Ordentlich and Shayevitz on unique-sum systems, we establish the above improved upper bound.
