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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 bystudying the extent which a random variable looks like a \emph{high entropy}distribution. There are different formal definitions of computational entropywith different advantages for different applications. Because of this, it is ofinterest to understand when these definitions are equivalent.  We consider three notions of computational entropy which are known to beequivalent when the test class $\cal{F}$ is closed under taking majorities.This equivalence constitutes (essentially) the so-called \emph{dense modeltheorem} of Green and Tao (and later made explicit by Tao-Zeigler, Reingold etal., and Gowers). The dense model theorem plays a key role in Green and Tao'sproof that the primes contain arbitrarily long arithmetic progressions and hassince been connected to a surprisingly wide range of topics in mathematics andcomputer 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 inturn provides examples where the dense model theorem is \emph{false}.

Comparing computational entropies below majority (or: When is the dense model theorem false?)