Arbitrarily slow decay in the logarithmically averaged Sarnak conjecture (Q6607358)

For a topological dynamical system \((X,T)\) (meaning that \(X\) is a compact metric space and \(T\colon X\to X\) a homeomorphism) the profound conjecture of \textit{P. Sarnak} [Not. South Afr. Math. Soc. 43, No. 2, 89--97 (2012; Zbl 1473.11147)] asks if \[\frac{1}{N}\sum_{n=1}^{N}\mu(n)f(T^nx)\rightarrow0\] as \(N\to\infty\) for any \(f\in C(X)\) and \(x\in X\) if \(T\) has zero topological dynamics. If \(X\) is a finite set this is equivalent to the prime number theorem in arithmetic progressions; if \(T\) is a rotation on the circle this is equivalent to Davenport's theorem. \N\NMany special classes of topological dynamical systems for which the conjecture holds have been found, but the general case seems remote. \textit{T. Tao} [Forum Math. Pi 4, Paper No. e8, 36 p. (2016; Zbl 1383.11116); Number theory -- Diophantine problems, uniform distribution and applications. Cham: Springer, 391--421 (2017); in: Number theory -- Diophantine problems, uniform distribution and applications. Festschrift in honour of Robert F. Tichy's 60th birthday. Cham: Springer. 391--421 (2017; Zbl 1426.11111)] introduced a logarithmic version of the conjecture, asking if \[\frac{\sum_{n=1}^{N}\mu(n)f(T^nx)/n}{\sum_{n=1}^{N}1/n} \rightarrow0\] for any \(f\in C(X)\) and \(x\in X\) as \(N\to\infty\) under the same hypothesis of zero entropy. \N\NHere examples of dynamical systems are constructed to show that the convergence to \(0\) can happen arbitrarily slowly, giving the logarithmically averaged version of earlier work by the authors [Ergodic Theory Dyn. Syst. 43, No. 9, 2863--2880 (2023; Zbl 1526.37022)], exhibiting arbitrarily slow convergence to \(0\) in the original conjecture.