Oscillating sequences, Gowers norms and Sarnak's conjecture

It is shown that there is an oscillating sequence of higher order which is not orthogonal to the class of dynamical flow with topological entropy zero. We further establish that any oscillating sequence of order $d$ is orthogonal to any $d$-nilsequence arising from the skew product on the $d$-dimensional torus $\mathbb{T}^d$. The proof yields that any oscillating sequence of higher order is orthogonal to any dynamical sequence arising from topological dynamical systems with quasi-discrete spectrum. however, we provide an example of oscillating sequence of higher order with large Gowers norms. We further obtain a new estimation of the average of M\"{o}bius function on the short interval by appealing to Bourgain's double recurence argument.