最硬的半空间

摘要：

研究了半空间$h:\{0,1\}^n\to\{0,1\}$在无穷范数下的多项式和有理函数的逼近问题。我们的主要结果是一个明确的建设“最难”的半空间，我们证明多项式和有理逼近下界匹配平凡的上界实现所有半空间。这完成了Myhill和Kautz（1961）开始的一系列漫长的工作。作为一个应用程序，我们构建了一个通信问题，基本上实现了最大可能的分离，$O(n)$与$2^{-Ω(n)},$之间的符号秩和差异。同样，我们的问题表现出的差距$\log n$与$Ω(n)$之间的通信复杂性与无界与弱无界的错误，提高二次在以前的建设和完成一系列的工作开始的Babai，弗兰克尔和西蒙（FOCS 1986）。我们的研究结果进一步推广到$k$ -党数的额头上的模型，在那里我们得到了明确的分离$\log n$与$Ω(n/4^{n})$的通信与无界与弱无界错误。这个差距是对以前工作的二次改进，并且与前额上数字下限的最新技术相匹配。

英文摘要：

We study the approximation of halfspaces $h:\{0,1\}^n\to\{0,1\}$ in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the "hardest" halfspace, for which we prove polynomial and rational approximation lower bounds that match the trivial upper bounds achievable for all halfspaces. This completes a lengthy line of work started by Myhill and Kautz (1961). As an application, we construct a communication problem that achieves essentially the largest possible separation, of $O(n)$ versus $2^{-Ω(n)},$ between the sign-rank and discrepancy. Equivalently, our problem exhibits a gap of $\log n$ versus $Ω(n)$ between the communication complexity with unbounded versus weakly unbounded error, improving quadratically on previous constructions and completing a line of work started by Babai, Frankl, and Simon (FOCS 1986). Our results further generalize to the $k$-party number-on-the-forehead model, where we obtain an explicit separation of $\log n$ versus $Ω(n/4^{n})$ for communication with unbounded versus weakly unbounded error. This gap is a quadratic improvement on previous work and matches the state of the art for number-on-the-forehead lower bounds.

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