  We show that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero average
along every infinite arithmetic progression then for every $N\ge 2$ there exist
(unilateral or bilateral) subshifts $\Sigma$ over $N$ symbols, with entropy
arbitrarily close to $\log N$, uncorrelated to $y$. In particular, for $y=\mu$
being the M\"obius function, we get that there exist subshifts as above which
satisfy the assertion of Sarnak's conjecture. The existence of positive entropy
systems uncorrelated to the M\"obius function is claimed in Sarnak's survey
\cite{sarnak} (and attributed to Bourgain), however, to our knowledge no
examples have ever been published. We fill in this gap and by the way we show
that this has nothing to do with more advanced algebraic properties (for
instance multiplicativity) of the considered sequence.
