A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Möbius function
Volume 251 / 2020
Abstract
A recent result of Downarowicz and Serafin (2019) shows that there exist positive entropy subshifts satisfying the assertion of Sarnak’s conjecture. More precisely, it is proved that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression (the Möbius function is an example of such a sequence $y$) then for every $N\ge 2$ there exists a subshift $\varSigma $ over $N$ symbols, with entropy arbitrarily close to $\log N$, uncorrelated to $y$.
In the present note, we improve the above result. First of all, we observe that the uncorrelation is uniform, i.e., for any continuous function $f:\varSigma \to \mathbb R $ and every $\epsilon \gt 0$ there exists $n_0$ such that for any $n\ge n_0$ and any $x\in \varSigma $ we have $$ \bigg |\frac 1n\sum _{i=1}^{n}f(T^ix)\,y_i\bigg | \lt \epsilon . $$ More importantly, by a fine-tuned modification of the construction we create a strictly ergodic subshift with the desired properties (uniformly uncorrelated to $y$ and with entropy arbitrarily close to $\log N$).
The question about these two additional properties (uniformity of uncorrelation and strict ergodicity) has been posed by Mariusz Lemańczyk in the context of the so-called strong MOMO (Möbius Orthogonality on Moving Orbits) property. Our result shows, among other things, that strong MOMO is essentially stronger than uniform uncorrelation, even for strictly ergodic systems.
