Diophantine approximation on circles and spheres

Diophantine approximation in number theory is to approximate a given irrational number with rational numbers. From a geometric perspective, it quantifies the rate at which a given geodesic flow approaches the cusp on the fundamental domain of the modular group within hyperbolic space. In this talk, we will discuss various Diophantine approximations on the real line and also on the complex plane. In the context of the complex plane, we approximate a complex number using elements from a specified imaginary quadratic field. Lastly, we will consider Diophantine approximation on circles and spheres and study intriguing examples.