Extremal values of continuants and transcendence of certain continued fractions

Abstract

We prove a criterion for the transcendence of continued fractions whose partial quotients are contained in a finite set { b 1,…, b r } of positive integers such that the density of occurrences of b i in the sequence of partial quotients exists for 1⩽ i⩽ r. As an application we study continued fractions [0, a 1, a 2, a 3,…] with a n =1+([ nθ]mod d) where θ is irrational and d⩾2 is a positive integer.