Three subjects of the Jacobi-Perron algorithm of dimension 2

TL;DR

This paper investigates the Jacobi-Perron algorithm of dimension 2, focusing on ideal convergence, algebraic conjugates, and the measure of bounded digit expansions, providing new conditions and generalizations.

Contribution

It introduces new sufficient conditions for ideal convergence, analyzes algebraic conjugates, and proves the null measure of bounded digit expansions in the algorithm.

Findings

Established conditions for ideal convergence of the algorithm.

Analyzed the behavior of algebraic conjugates in relation to continued fractions.

Proved that the set of points with bounded digits has measure zero.

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].