Three subjects of the Jacobi-Perron algorithm of dimension 2

Abstract

We shall study three subjects of the Jacobi-Perron Algorithm of dimension 2. First, we study the "ideal convergence". About the approximations (p_n/r_n, q_n/r_n) to (A, B) (where A and B are positive real numbers, r_n, p_n and q_n are natural numbers), Showing some inequalities and evaluations of |p_n-Ar_n| and |q_n-Br_n|, we shall prove some sufficient conditions for p_n-Ar_n and q_n-Br_n converge at 0, in which case the algorithm is said to be ideally convergent(see [1]). Second, in connection with the classical continued fractions, we treat the behavior of the algebraic conjugates. Third, we shall prove that the set of (A, B) for which the digits of expansions are bounded from above is null which is a generalization of Theorem 196 in [3].