Arithmetics on number systems with irrational bases

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.