Metric number theory of Fourier coefficients of Hilbert modular forms

In this note, we attempt to generalize the work of Bengoechea (2019) to the case of Hilbert cusp forms. We prove Diophantine approximation of a real number by Fourier coefficients (supported at prime powers) of a primitive Hilbert cusp forms and we compute the density of such prime ideals. We also study the refined rate of approximation of real numbers by Fourier coefficients of primitive Hilbert cusp forms, using techniques and results from metric number theory such as Schmidt's game and metrical inhomogeneous Diophantine approximation.