Translator Disclaimer
December 2003 Arithmetics on number systems with irrational bases
P. Ambrož, C. Frougny, Z. Masáková, E. Pelantová
Bull. Belg. Math. Soc. Simon Stevin 10(5): 641-659 (December 2003). DOI: 10.36045/bbms/1074791323

Abstract

For irrational $\beta>1$ we consider the set ${\rm Fin}(\beta)$ of real numbers for which $|x|$ has a finite number of non-zero digits in its expansion in base $\beta$. In particular, we consider the set of $\beta$-integers, i.e. numbers whose $\beta$-expansion is of the form $\sum_{i=0}^nx_i\beta^i$, $n\geq0$. We discuss some necessary and some sufficient conditions for ${\rm Fin(\beta)}$ to be a ring. We also describe methods to estimate the number of fractional digits that appear by addition or multiplication of $\beta$-integers. We apply these methods among others to the real solution $\beta$ of $x^3=x^2+x+1$, the so-called Tribonacci number. In this case we show that multiplication of arbitrary $\beta$-integers has a fractional part of length at most 5. We show an example of a $\beta$-integer $x$ such that $x\cdot x$ has the fractional part of length $4$. By that we improve the bound provided by Messaoudi from value 9 to 5; in the same time we refute the conjecture of Arnoux that 3 is the maximal number of fractional digits appearing in Tribonacci multiplication.

Citation

Download Citation

P. Ambrož. C. Frougny. Z. Masáková. E. Pelantová. "Arithmetics on number systems with irrational bases." Bull. Belg. Math. Soc. Simon Stevin 10 (5) 641 - 659, December 2003. https://doi.org/10.36045/bbms/1074791323

Information

Published: December 2003
First available in Project Euclid: 22 January 2004

zbMATH: 1126.11308
MathSciNet: MR2073762
Digital Object Identifier: 10.36045/bbms/1074791323

Subjects:
Primary: 11A67, 68Q42

Rights: Copyright © 2003 The Belgian Mathematical Society

JOURNAL ARTICLE
19 PAGES


SHARE
Vol.10 • No. 5 • December 2003
Back to Top