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I am reading the paper Sign Changes in Hecke Eigenvalues by Matomaki and Radziwill, and in one place they mention the following,

It would be interesting to rule out the possibility of $\lambda_f(n)$ regularly flipping its sign and thus to investigate the entropy of the sequence of sign of $\lambda_f(n)$.

My Question : What is the meaning of entropy of sign of $\lambda_f(n)$?

The only connection that I can make is the following. There is a notion of Nilsequences and nilmanifolds where you consider sequences arising from the orbit of a point in a (topological) dynamical system along with a continuous function. Then I guess it makes sense to consider the entropy of that system and somehow use it to define the entropy of a sequence, but I don't see how that features here.

Any help is greatly appreciated.

P.S. I am still in the process of learning about Nilsequences and Nilmanifolds (particularly the work of Green and Tao on the Mobius function). Answers that assume minimal amount of this theory will be greatly uselful.

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    $\begingroup$ Let $(\sigma_n)_{n=1}^\infty \subseteq \{-1,+1\}$ be a sequence. Define $f(k) := \#\{(\epsilon_1,\dots,\epsilon_k) \in \{-1,+1\}^k : \exists n \in \mathbb{N} \hspace{1mm} \text{ s.t. } \epsilon_i = \sigma_{n+i} \hspace{1.5mm} \forall 1 \le i \le k\}$. Then maybe they mean $\lim_{k \to \infty} \frac{1}{k}\log_2(f(k))$. I'm falling asleep so idk. $\endgroup$
    – mathworker21
    Commented Aug 7, 2022 at 8:58
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    $\begingroup$ Mathworker21's definition is probably what's intended, and you are correct about a connection to dynamical systems. However, nilsystems are always zero entropy, and $\lambda_n(f)$ is (I assume) conjectured to have positive entropy. If so, the associated dynamical system will probably have a Bernoulli factor - see Theorem 1.6 of of this article by Frantzikinakis and Host for explicit examples of constructing and analyzing systems associated to number theoretic sequences. $\endgroup$
    – John Griesmer
    Commented Aug 7, 2022 at 17:43

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