Arithmetics on number systems with irrational bases

For irrational β > 1 we consider the set Fin(β) of real numbers for which |x| has a finite number of non-zero digits in its expansion in base β. In particular, we consider the set of β-integers, i.e. numbers whose β-expansion is of the form ∑n i=0 xiβ , n ≥ 0. We discuss some necessary and some sufficient conditions for Fin(β) to be a ring. We also describe methods to estimate the number of fractional digits that appear by addition or multiplication of βintegers. We apply these methods among others to the real solution β of x3 = x2 + x+ 1, the so-called Tribonacci number. In this case we show that multiplication of arbitrary β-integers has a fractional part of length at most 5. We show an example of a β-integer x such that x ·x has the fractional part of length 4. By that we improve the bound provided by Messaoudi [11] 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.