We answer four questions from a recent paper of Rao and Shinkar [17] on Lipschitz bijections between functions from {0, 1}n to {0, 1}. (1) We show that there is no O(1)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on O(1) input bits. (2) We give a construction for a mapping from XOR to Majority which has average stretch $O(\sqrt{n})$ , matching a previously known lower bound. (3) We give a 3-Lipschitz embedding $\phi \colon \{0,1\}^n \to \{0,1\}^{2n+1}$ such that $${\rm{XOR }}(x) = {\rm{ Majority }}(\phi (x))$$ for all $x \in \{0,1\}^n$ . (4) We show that with high probability there is an O(1)-bi-Lipschitz mapping from Dictator to a uniformly random balanced function.

中文翻译：

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布尔函数之间的 Lipschitz 双射

我们回答了 Rao 和 Shinkar [17] 最近关于 {0, 1} 函数之间的 Lipschitz 双射的论文中的四个问题n到 {0, 1}。(1) 我们证明没有○(1)-bi-Lipschitz 双射从 Dictator 到 XOR，使得每个输出位取决于○(1) 输入位。(2) 我们给出了从 XOR 到具有平均拉伸的多数的映射的构造$O(\sqrt{n})$，匹配先前已知的下限。(3) 我们给出一个 3-Lipschitz 嵌入$\phi \冒号 \{0,1\}^n \to \{0,1\}^{2n+1}$这样$${\rm{XOR }}(x) = {\rm{ 多数 }}(\phi (x))$$对所有人$x \in \{0,1\}^n$. (4) 我们证明，很有可能存在○(1)-bi-Lipschitz 从独裁者映射到均匀随机平衡函数。