On an inhomogeneous uniform Littlewood type problem

We show that a fully inhomogeneous uniform Littlewood type problem has a negative answer and the counterexamples form a dense $G_{\delta }$ set. This extends the author’s recent analogous result for the homogeneous case. The main difficulty in the general setting is the semi-homogeneous case where one linear form is homogeneous and the other inhomogeneous with irrational shift. We further address the higher dimensional analogue.