On Convergents of Proper Continued Fractions

TL;DR

This paper investigates the approximation properties of proper continued fractions, classifies their convergents, and introduces a new dynamical system generalizing the Gauss map with proven ergodicity.

Contribution

It provides a complete classification of convergents for odd indices, near-complete for even indices, and proposes a novel ergodic dynamical system for generating continued fractions.

Findings

Classified possible convergents for odd index continued fractions.

Established ergodicity of a new two-dimensional Gauss map.

Reduced convergence analysis to indices one and two.

Abstract

Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their convergents and classify which pairs of integers $p,q$ yield a convergent $p/q$ to some irrational $x$. Notably, we reduce the problem to finding convergence only of index one and two. We completely classify the possible choices for convergents of odd index and provide a near-complete classification for even index. We furthermore propose a natural two-dimensional generalization of the classical Gauss map as a method for dynamically generating all possible expansions and establish ergodicity of this map.