Valerie Berthe ; Laurent Imbert - Diophantine Approximation, Ostrowski Numeration and the Double-Base Number System

dmtcs:450 - Discrete Mathematics & Theoretical Computer Science, March 1, 2009, Vol. 11 no. 1 - https://doi.org/10.46298/dmtcs.450
Diophantine Approximation, Ostrowski Numeration and the Double-Base Number SystemArticle

Authors: Valerie Berthe ORCID1; Laurent Imbert ORCID2,1

  • 1 Arithmétique informatique
  • 2 Pacific Institute for the Mathematical Sciences

A partition of $x > 0$ of the form $x = \sum_i 2^{a_i}3^{b_i}$ with 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 \mbox{$\{2,3\}$-integer}, i.e., a number of the form $2^a3^b$, 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.


Volume: Vol. 11 no. 1
Section: Analysis of Algorithms
Published on: March 1, 2009
Imported on: March 26, 2015
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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