Computational pseudorandomness studies the extent to which a random variable $\bf{Z}$ looks like the uniform distribution according to a class of tests $\cal{F}$. Computational entropy generalizes computational pseudorandomness by studying the extent which a random variable looks like a \emph{high entropy} distribution. There are different formal definitions of computational entropy with different advantages for different applications. Because of this, it is of interest to understand when these definitions are equivalent. We consider three notions of computational entropy which are known to be equivalent when the test class $\cal{F}$ is closed under taking majorities. This equivalence constitutes (essentially) the so-called \emph{dense model theorem} of Green and Tao (and later made explicit by Tao-Zeigler, Reingold et al., and Gowers). The dense model theorem plays a key role in Green and Tao's proof that the primes contain arbitrarily long arithmetic progressions and has since been connected to a surprisingly wide range of topics in mathematics and computer science, including cryptography, computational complexity, combinatorics and machine learning. We show that, in different situations where $\cal{F}$ is \emph{not} closed under majority, this equivalence fails. This in turn provides examples where the dense model theorem is \emph{false}.

中文翻译：

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比较低于多数的计算熵（或：密集模型定理何时是错误的？）

计算伪随机性研究随机变量 $\bf{Z}$ 根据一类测试 $\cal{F}$ 看起来像均匀分布的程度。计算熵通过研究随机变量看起来像 \emph {高熵} 分布的程度来概括计算伪随机性。计算熵有不同的形式定义，对于不同的应用具有不同的优势。因此，了解这些定义何时等效是很有意义的。我们考虑了计算熵的三个概念，当测试类 $\cal{F}$ 在大多数情况下关闭时，它们是等价的。这种等价（本质上）构成了格林和道的所谓\emph{密集模型定理}（后来由陶-齐格勒、雷因戈尔德等人和高尔斯明确提出）。密集模型定理在 Green 和 Tao 证明质数包含任意长的算术级数的证明中发挥了关键作用，并且此后与数学和计算机科学中令人惊讶的广泛主题相关联，包括密码学、计算复杂性、组合学和机器学习。我们表明，在 $\cal{F}$ 在多数情况下 \emph{not} 关闭的不同情况下，这种等价性失败。这反过来提供了密集模型定理为 \emph{false} 的示例。在 $\cal{F}$ 在多数情况下 \emph{not} 关闭的不同情况下，这种等价性失败。这反过来提供了密集模型定理为 \emph{false} 的示例。在 $\cal{F}$ 在多数情况下 \emph{not} 关闭的不同情况下，这种等价性失败。这反过来提供了密集模型定理为 \emph{false} 的示例。