Metric number theory of Fourier coefficients of Hilbert modular forms

Key Points

• The research aims to generalize existing work on Hilbert cusp forms focusing on Diophantine approximation of real numbers.
• Proved Diophantine approximation for real numbers by Fourier coefficients of primitive Hilbert cusp forms.
• Computed the density of prime ideals related to these forms.
• Utilized techniques from metric number theory, including Schmidt's game.
• Demonstrated efficient approximation of real numbers by Fourier coefficients supported at prime powers.
• Established a refined rate of approximation linked to the density of prime ideals.
• Confirmed relationships between Diophantine approximation and properties of Hilbert cusp forms.