Diophantine approximation, ostrowski numeration and the double-base number system

Abstract

A partition of x > 0 of the form x = Σi2ai3biwith distinct parts is called a double-base expansion of x. Such a representation can be obtained using a greedy approach, assuming one can efficiently compute the largest {2,3}-integer, i.e., a number of the form 2a3b, less than or equal to x. In order to solve this problem, we propose an algorithm based on continued fractions in the vein of the Ostrowski number system, we prove its correctness and we analyse its complexity. In a second part, we present some experimental results on the length of double-base expansions when only a few iterations of our algorithm are performed. © 2009 Discrete Mathematics and Theoretical Computer Science (DMTCS),.