Year: 2026
Type: article
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 x. 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.
Authors: Niels Langeveld, David Ralston
Institutions: Montanuniversität Leoben, SUNY Old Westbury
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4405174056