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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] Polynomial Sarnak 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).
            • Remark. [A. Kanigowski, M. Lemańczyk, M. Radziwill] A counterexample to Polynomial Sarnak Conjecture 1 has recently been given in “Prime Number Theorem for Analytic Skew Products” by A. Kanigowski, M. Lemańczyk and M. Radziwill in the class of strictly ergodic (i.e. minimal and uniquely ergodic) continuous Anzai skew products. Independently and simultaneously, a counterexample has also been given by Z. Liang and R. Shi in “A counter-example for polynomial Sarnak Conjecture” in the class of minimal Toeplitz subshifts.
                • 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.
                    1. Remark. [Thierry de la Rue] In this question we assume also that $\sum_j 1/a_j =\infty$, otherwise the conclusion does not hold.
                        • 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.