On convergents of proper continued fractions
Abstract
Proper continued fractions are generalized continued fractions with positive integer numerators [Formula: see text] and integer denominators with [Formula: see text]. In this paper, we study the strength of approximation of numbers to their convergents and classify which pairs of integers [Formula: see text] yield a convergent [Formula: see text] to some [Formula: see text]. Notably, we reduce the problem to finding convergents only of index one and two, both of which we completely classify. 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.
Fields of Study & Subjects
Approximation Theory and Sequence Spaces
Mathematical and Theoretical Analysis
Publication details
- Journal / source
- International Journal of Number Theory
- Publisher
- World Scientific Pub Co Pte Ltd
- Volume
- 22
- Issue
- 10
- Pages
- 1–25
- Publication date
- 12 June 2026
- Language
- EN