Arithmetics on number systems with irrational bases

For irrational beta > 1 we consider the set 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 Sigma(i=0)(n) x(i)beta(i), n greater than or equal to 0. We discuss some necessary and some sufficient conditions for 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 (.) x has the fractional part of length 4. By that we improve the bound provided by Messaoudi [12] 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.