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Register a new userPolynomial mean complexity and Logarithmic Sarnak conjecture
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In this paper, we reduce the logarithmic Sarnak conjecture to the ${0,1}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.
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In this paper, we show that for any sequence ${bf a}=(a_n)_{nin Z}in {1,ldots,k}^mathbb{Z}$ and any $epsilon>0$, there exists a Toeplitz sequence ${bf b}=(b_n)_{nin Z}in {1,ldots,k}^mathbb{Z}$ such that the entropy $h({bf b})leq 2 h({bf a})$ and $lim_{Ntoinfty}frac{1}{2N+1}sum_{n=-N}^N|a_n-b_n|<epsilon$. As an application of this result, we reduce Sarnak Conjecture to Toeplitz systems, that is, if the M{o}bius function is disjoint from any Toeplitz sequence with zero entropy, then the Sarnak conjecture holds.
We construct the counter-example for polynomial version of Sarnaks conjecture for minimal systems, which assets that the Mobius function is linearly disjoint from subsequences along polynomials of deterministic sequences realized in minimal systems. Our example is in the class of Toeplitz systems, which are minimal.
In this paper we study $C^*$-algebra version of Sarnak Conjecture for noncommutative toral automorphisms. Let $A_Theta$ be a noncommutative torus and $alpha_Theta$ be the noncommutative toral automorphism arising from a matrix $Sin GL(d,mathbb{Z})$. We show that if the Voiculescu-Brown entropy of $alpha_{Theta}$ is zero, then the sequence ${rho(alpha_{Theta}^nu)}_{nin mathbb{Z}}$ is a sum of a nilsequence and a zero-density-sequence, where $uin A_Theta$ and $rho$ is any state on $A_Theta$. Then by a result of Green and Tao, this sequence is linearly disjoint from the Mobius function.
We study dynamical systems that have bounded complexity with respect to three kinds of directional metrics: the directional Bowen metric $d_k^{vec{v},b}$, the directional max-mean metric $hat{d}_k^{vec{v},b}$ and the directional mean metric $overline{d}_k^{vec{v},b}$. It is shown that a $mathbb{Z}^q$-topological system $(X,T)$ has bounded topological complexity with respect to ${d_k^{vec{v},b}}$ (respectively ${hat{d}_k^{vec{v},b}}$) if and only if $T$ is $(vec{v},b)$-equicontinuous (respectively $(vec{v},b)$-equicontinuous in the mean). It turns out that a measure $mu$ has bounded complexity with respect to ${d_k^{vec{v},b}}$ if and only if $T$ is $(mu,vec{v},b)$-equicontinuous. Whats more, it is shown that $mu$ has bounded complexity with respect to ${overline{d}_k^{vec{v},b}}$ if and only if $mu$ has bounded complexity with respect to ${hat{d}_k^{vec{v},b}}$ if and only if $T$ is $(mu,vec{v},b)$-mean equicontinuous if and only if $T$ is $(mu,vec{v},b)$-equicontinuous in the mean if and only if $mu$ has $vec{v}$-discrete spectrum.
Assuming Schanuels conjecture, we prove that any polynomial exponential equation in one variable must have a solution that is transcendental over a given finitely generated field. With the help of some recent results in Diophantine geometry, we obtain the result by proving (unconditionally) that certain polynomial exponential equations have only finitely many rational solutions. This answers affirmatively a question of David Marker, who asked, and proved in the case of algebraic coefficients, whether at least the one-variable case of Zilbers strong exponential-algebraic closedness conjecture can be reduced to Schanuels conjecture.
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