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.
@article{1074791323,
author = {Ambro\v z, P. and Frougny, C. and Mas\'akov\'a, Z. and Pelantov\'a, E.},
title = {Arithmetics on number systems with irrational bases},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {10},
number = {1},
year = {2003},
pages = { 641-659},
language = {en},
url = {http://dml.mathdoc.fr/item/1074791323}
}
Ambrož, P.; Frougny, C.; Masáková, Z.; Pelantová, E. Arithmetics on number systems with irrational bases. Bull. Belg. Math. Soc. Simon Stevin, Tome 10 (2003) no. 1, pp. 641-659. http://gdmltest.u-ga.fr/item/1074791323/