A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Moebius function

A recent result of Downarowicz and Serafin (DS) 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 M\"obius function is an example of such a \sq\ $y$) then for every $N\ge 2$ there exists a subshift $\Sigma$ over $N$ symbols, with entropy arbitrarily close to $\log N$, uncorrelated to $y$. In the present note, we improve the result of (DS). First of all, we observe that the uncorrelation obtained in (DS) is \emph{uniform}, i.e., for any continuous function $f:\Sigma\to {\mathbb R}$ and every $\epsilon>0$ there exists $n_0$ such that for any $n\ge n_0$ and any $x\in\Sigma$ we have $$\left|\frac1n\sum_{i=1}^{n}f(T^ix)\,y_i\right|<\epsilon.$$ More importantly, by a fine-tuned modification of the construction from (DS) we create a \emph{strictly ergodic} subshift, with all the desired properties of the example in (DS) (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 Lemanczyk in the context of the so-called strong MOMO (M\"obius 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.