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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 $\sum_{i=0}^nx_i\beta^i$. 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 $\beta$ solution 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.
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