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3. Möbius and Liouville systems


    1. Add remark

      Problem 3.1.

      Can a Liouville system be isomorphic to $(\mathbb{T}^2, m_{\mathbb{T}^2}, T)$, where $T:\mathbb{T}^2\to \mathbb{T}^2$ is given by $\left(\begin{smallmatrix} x\\ y \end{smallmatrix}\right) \mapsto \left(\begin{smallmatrix} x+y\\ y \end{smallmatrix}\right)$ mod $1$?
        1. Remark. [M. Lemańczyk] This unipotent system appears as a Furstenberg system of some aperiodic multiplicative functions (see A. Gomilko, M. Lemańczyk, T. de la Rue, "On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł and Tao" in J. Modern Dynamics 17 (2021)).


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              Problem 3.2.

              Is $-1$ in the spectrum of a Liouville system? Equivalently, can we show that \begin{equation*} \sum_{n\leq X} (-1)^{n-1} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o(\log X)? \end{equation*}
                1. Remark. [Nikos] More generally, one can ask to show that $e(p/q)$ is not in the spectrum of a Liouville system when $p/q$ is not an integer. If this is shown, one immediate consequence (using Szemerédi’s theorem) is that on the range of the Liouville function there are arbitrarily large patterns $++\ldots+$ and $--\ldots-$ (see Problem 2.2).


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                      Problem 3.3.

                      Let $X_{\mu^2} = \overline{\{T^k(\mu^2) : k \in \N\}}$ and $X_{\mu} = \overline{\{T^k(\mu) : k \in \N\}}$, for $T$ the shift map. What are the pre-images of $f:X_{\mu} \to X_{\mu^2}$ given by $(x_i) \mapsto (x_i^2)$ in $X_{\mu^2}$?

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