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7. Generalizations

    1. Problem 7.1.

      We have the following polynomial versions of the Sarnak and Chowla conjectures, where $(X, \mu, T)$ is a $0$-entropy system, $f \in C(X)$, $x \in X$ and $p(t) \in \Z[t]$:

      • Polynomial Sarnak Conjecture 1: If $T$ is minimal, then $\sum_{n\leq X} \lambda(n) f(T^{p(n)}x) = o(X)$.
      • Polynomial Sarnak Conjecture 2: If $p$ is not a constant multiple of the square of a polynomial, then $\sum_{n \leq X} \lambda(p(n)) f(T^nx) = o(X)$.
      • Polynomial Chowla Conjecture: If $p$ is not a constant multiple of the square of a polynomial, then $\sum_{n \leq X} \lambda(p(n)) = o(X)$.

      Is polynomial Sarnak true, and does it follow from polynomial Chowla in any of the two cases?
        1. Remark. [Nikos] In Conjecture 2 write that $p$ is a non-constant multiple of the square of a polynomial [Done, thanks!]. This corrected variant of Conjecture 2 follows from the Polynomial Chowla conjecture using the exact same argument that shows that the Chowla conjecture for $\lambda(n)$ implies the Sarnak Conjecture for $\lambda(n)$ (this argument applies to any bounded sequence $a(n)$ with vanishing correlations, in particular it applies to $a(n)=\lambda(p(n))$ if we assume that the polynomial Chowla conjecture holds).
            • Problem 7.2.

              Let $\mathcal{A} = \{a_1, a_2, \dots\}$ be a sequence of integers with $(a_i, a_j) = 1$ for $i \not= j$. Let $\pi(n):= (-1)^{\#\{a_j:a_j|n\}}$. Is it true that \begin{equation*} \sum_{n \leq X} \pi(n) = o(X)? \end{equation*} Can we find a set $\mathcal{A}$ such that \begin{equation*} \sum_{n \leq X} \pi(n+h_1) \dots \pi(n+h_k) = o(X) \end{equation*} for all distinct $h_1, \dots, h_k$?

              • Remark: Let \[ \lambda_y(p) = \begin{cases} -1, &\text{if } p \leq y \\ +1, &\text{if } p > y. \end{cases} \] If $y = X^{o(1)}$ then \begin{equation*} \sum_{n\leq X} \lambda_y(n+h_1) \cdots \lambda_y(n+h_k) = o(X). \end{equation*} For $k=2$ this was done by Daboussi-Sarkar.
                • Problem 7.3.

                  Suppose that for all fixed $\chi$ mod $q$ and $t \in \R$, $f:\N \to \C$ is multiplicative with \begin{equation*} \sum_{p} \frac{1 - \operatorname{Re} f(p) \overline{\chi(p) p^{it}}}{p} = \infty. \end{equation*} Does there exists a subsequence $X_k \to \infty$ such that \begin{equation*} \sum_{n \leq X_j} f(n) \overline{f(n+1)} = o(X_j)? \end{equation*} Alternatively, is \begin{equation*} \sum_{n\leq X} f(n) f(n+h_1) \cdots f(n+h_k) = o(X) \end{equation*} for distinct $h_1, \dots, h_k$?
                    1. Remark. [Nikos] On the second statement, regarding higher order correlations, one should ask whether the conclusion holds for some fixed subsequence $X_j\to \infty$ (independent of the $k$ and $h_1, \ldots, h_k$) and also allow some of the $f$’s to be equal to the conjugate of $f$. Moreover, one would also be happy with a result about logarithmic averages.
                        • Problem 7.4.

                          What is the Möbius disjointness variant of the Jewett-Krieger theorem?

                              Cite this as: AimPL: Sarnak's conjecture, available at http://aimpl.org/sarnakconjecture.