The relative growth rate for partial quotients

TL;DR

This paper investigates how the partial quotients of irrational numbers grow in relation to their approximation by convergents, analyzing exceptional sets with computed Hausdorff dimensions.

Contribution

It introduces a new analysis of the growth rate of partial quotients relative to approximation, including Hausdorff dimension calculations for exceptional cases.

Findings

Hausdorff dimension of certain exceptional sets computed

Growth rate of partial quotients characterized in non-generic cases

Relation between partial quotients and approximation rate established

Abstract

We look at the rate of growth of the partial quotients of the infinite continued fraction expansion of an irrational number relative to the rate of approximation of the number by its convergents. In non-generic cases the Hausdorff dimension of some exceptional sets is computed.