M\"{o}bius Disjointness for product flows of rigid dynamical systems and affine linear flows

We obtain that Sarnak's M\"{o}bius Disjointness Conjecture holds for product flows between affine linear flows on compact abelian groups of zero topological entropy and a class of rigid dynamical systems. To prove this, we show an estimate for the average value of the product of the M\"obius function and any polynomial phase over short intervals and arithmetic progressions simultaneously. In addition, we prove that the logarithmically averaged M\"obius Disjointness Conjecture holds for the product flow between any affine linear flow on a compact abelian group of zero entropy and any rigid dynamical system. As an application, we show that the logarithmically averaged M\"obius Disjointness Conjecture holds for every Lipschitz continuous skew product dynamical system on $\mathbb{T}^2$ over a rotation of the circle.