Usually, when we are looking for approximations of the irrational \alpha by rational fractions p/q, we want to solve the system of inequalities
|\alpha q - p|< \varepsilon, 1\le q \le Q
in integers p,q.
This formulation of the problem (corresponding to the L_\infty-norm) leads to ordinary continued fractions.
Similar formulations corresponding to the L_2 and L_1-norms go back to Hermite and Minkowski. They are related to other (irregular) continued fraction expansion algorithms. We will discuss these algorithms and explain how
these constructions are related to the relatively new concept of Dirichlet improvability.
This is a joint work with Nikita Shulga.