Oppenheim continued fraction expansion and Beta-expansion

Abstract

For β>1 and x∈[0,1), let kn(x) represent the exact number of digits in the Oppenheim continued fraction expansions of x given by the first n digits in the β-expansion of x. In this paper, we obtain the relations between kn(x) and n as limsupn→∞lnn=0, where ln=sup⁡{k≥0:ϵn+j⁎(1)=0,1≤j≤k} and 1=∑n=1∞ϵn⁎(1)βn is the infinite β-expansion of 1.