A singular function from Sturmian continued fractions

Abstract

. For α ≥ 1, let s α ( n ) = ⌈ αn ⌉ − ⌈ α ( n − 1) ⌉ . A continued fraction C ( α ) = [0; s α (1) ,s α (2) , ... ] is considered and analyzed. Appeal- ing to Diophantine approximation, we investigate the differentiability of C ( α ), and then show its singularity.