Sarnak's conjecture asserts that any dynamical system with zero topological entropy is uncorrelated to the Möbius function. While the problem is far from being resolved, the reversed question seems a bit easier: is every dynamical system with positive topological entropy correlated to the Möbius function? In his Three lectures Peter Sarnak answers this question (negatively), attributing appropriate examples to Bourgain. Probably this fact has stopped other people from further attempts to construct such examples, while Bourgain actually never published his, and until now the claim is only to be found in Sarnak's Three lectures without any solid evidence. It is not even clear whether Bourgain still supports his claim.

With Jacek Serafin we have decided to fill in this gap. We construct binary subshifts with entropy arbitrarily close to log 2, which are uncorrelated to the Möbius function. In fact, the Möbius function is completely inessential in the argument. It can be replaced by any (complex) bounded sequence with zero average. A key tool in the construction is the elementary Hoeffding's inequality.