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 \(2^{-\Omega(n)}\), between the sign-rank and discrepancy. Equivalently, our problem exhibits a gap of log n versus \(\Omega(n)\) 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 \(\Omega(n/4^{n})\) for communication with unbounded versus weakly unbounded error.

我们通过任意给定次数的多项式和有理函数来研究无穷范数中半空间\(h:\{0,1\}^n\to\{0,1\}\)的近似。我们的主要结果是“最难”半空间的显式构造，为此我们证明了与所有半空间可实现的平凡上限相匹配的多项式和有理逼近下界。这完成了由 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，该问题本质上实现了符号秩和差异之间O(n)与\(2^{-\Omega(n)}\)的最大可能分离。同样，我们的问题在无界误差与弱无界误差的通信复杂性之间表现出 log n与\(\Omega(n)\)之间的差距，在以前的构造上进行了二次改进，并完成了 Babai、Frankl 和 Simon 开始的一系列工作（FOCS 1986）。我们的结果进一步推广到k方额头上的数字模型，在该模型中，我们获得了 log n与\(\Omega(n/4^{n})\)的显式分离 ，以用于无界与弱无界的通信错误。