Weakly tame systems, their characterizations and applications

Abstract

We explore the notion of discrete spectrum and its various characterizations for ergodic measure-preserving actions of an amenable group on a compact metric space. We introduce a notion of ‘weak-tameness’, which is a measure theoretic version of a notion of ‘tameness’ introduced by E. Glasner, based on the work of A. Köhler, and characterize such topological dynamical systems as systems for which every invariant measure has a discrete spectrum. Using the work of M. Talagrand, we also characterize weakly tame as well as tame systems in terms of the notion of ‘witness of irregularity’ which is based on ‘up-crossings’. Then we establish that ‘strong Veech systems’ are tame. In particular, for any countable amenable group T, the flow on the orbit closure of the translates of a ‘Veech function’ $f \in {\mathbb {K}}(T)$ is tame. Thus Sarnak’s Möbius orthogonality conjecture holds for this flow and as a consequence, we obtain an improvement of Motohashi–Ramachandra theorem of 1976 on the Mertens function in short intervals. We further improve Motohashi–Ramachandra’s bound to 1/2 under the Chowla conjecture.

Appendix. On the average Chowla of order two

In this short note, by applying a observation of Bourgain observation, we present a simple proof of Matomäki-Radziwiłł-Tao theorem on the average Chowla of order two [33], based on Davenport theorem. In their inequality H is allowed to grow very slowly with respect to X. Here, for $H=X$ we obtain a bound for the speed of convergence. Notice that this is the only ingredient needed for the proof of the validity of Sarnak conjecture for systems with discrete spectrum in Huang-Wang-Ye’s result.

Theorem .12

Let $\varvec{\nu }$ be a Möbius or Liouville function. Then, for any $N \ge 2$,

$$\begin{aligned}&\frac{1}{N}\sum _{m=1}^{N}\Big |\frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n) \varvec{\nu }(n+m)\Big | \le \frac{C}{\log (N)^{\kappa }}. \end{aligned}$$

where C is some positive constant.

Proof

By Cauchy–Schwarz inequality, we have

$$\begin{aligned}&\frac{1}{N}\sum _{m=1}^{N}\Big |\frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n) \varvec{\nu }(n+m)\Big | \le \left( \frac{1}{N}\sum _{m=1}^{N}\left| \frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n) \varvec{\nu }(n+m)\right| ^2\right) ^{\frac{1}{2}}, \end{aligned}$$

and by Bourgain’s observation [7, equations (2.5) and (2.7)], we have

$$\begin{aligned}&\sum _{m=1}^{N}\left| \frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n)\varvec{\nu }(n+m) \lambda ^{n+m}\right| ^2 \nonumber \\&\quad = \sum _{m=1}^{N}\left| \int _{{\mathbb {T}}}\left( \frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n)z^{-n}\right) \left( \sum _{p=1}^{2N} \varvec{\nu }(p) {\left( \lambda z\right) }^p\right) z^{-m} dz\right| ^2 \nonumber \\&\quad \le \int _{{\mathbb {T}}}\left| \frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n) z^{-n}\right| ^2\left| \sum _{p=1}^{2N} \varvec{\nu }(p) {\big (\lambda z\big )}^p\right| ^2 dz\\&\quad \le \sup _{z \in {\mathbb {T}}}\left( \left| \frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n) z^{-n}\right| \right) ^2 \int _{{\mathbb {T}}}\left| \sum _{p=1}^{2N} \varvec{\nu }(p) {\big (\lambda z\big )}^p\right| ^2 dz. \end{aligned}$$

The inequality (1.3) is due to Parseval inequality. Indeed, by putting

$$\begin{aligned} \Phi _N(z)=\left( \frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n) z^{-n}\right) \left( \sum _{p=1}^{2N} \varvec{\nu }(p) {\big (\lambda z\big )}^p \right) . \end{aligned}$$

We see that for any $m \in {\mathbb {Z}}$,

$$\begin{aligned} \widehat{\Phi _N}(m) =\int _{{\mathbb {T}}}\left( \frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n)z^{-n}\right) \left( \sum _{p=1}^{2N} \varvec{\nu }(p) {\big (\lambda .z\big )}^p\right) z^{-m} dz. \end{aligned}$$

and

$$\begin{aligned} \sum _{m=1}^{N}\left| \int _{{\mathbb {T}}}\left( \frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n) z^{-n}\right) \left( \sum _{p=1}^{2N} \varvec{\nu }(p) {\big (\lambda .z\big )}^p\right) z^{-m} dz\right| ^2= & {} \sum _{m=1}^{N}\left| \widehat{\Phi _N}(m)\right| ^2\\\le & {} \int _{{\mathbb {T}}} |\Phi _N(z)|^2 dz. \end{aligned}$$

Now, by appealing to Davenport Theorem, we get

$$\begin{aligned}&\sum _{m=1}^{N}\left| \frac{1}{N}\sum _{n=1}^{N} \varvec{\nu }(n)\varvec{\nu }(n+m) \lambda ^{n+m}\right| ^2 \le \frac{C_\epsilon }{\log (N)^{\epsilon }} \end{aligned}$$

From this, we obtain the desired inequality and the proof is complete.