Metrical properties for a class of continued fractions with increasing digits

In 2002, Hartono, Kraaikamp and Schweiger introduced the Engel continued fractions (ECF), whose partial quotients are increasing. Later, Schweiger generalized it into a class of continued fractions with increasing digits and a parameter ϵ, called generalized continued fractions (GCF). In this paper, we will give some arithmetic properties of such an expansion, and show that the GCF holds similar metric properties with ECF under the condition that −1<ϵ⩽1. But when it comes to the condition that ϵ=−1, it does not.