Finite beta-expansions. (English) Zbl 0814.68065

Let \(\beta>1\) be a real number and for a real number \(x\), \([x]\) be its integer part, \(\{x\}\) be its fractional part. There exists \(k \in \mathbb{Z}\) such that \(\beta^ k \leq x < \beta^{k+1}\). Let \(x_ k = [x/ \beta^ k]\), and \(r_ k = \{x/ \beta^ k\}\). Then for \(k > i \geq - \infty\), put \(x_ i = [\beta r_{i+1}]\), and \(r_ i = \{\beta r_{i+1}\}\). We get a \(\beta\)-expansion \(x = x_ k \beta^ k + x_{k-1} \beta^{k-1} + \cdots\). If \(k < 0(x < 1)\), we put \(x_ 0 = x_{-1} = \cdots + x_{k+1} = 0\) (the ‘greedy algorithm’). If an expansion ends in infinitely many zeros, it is said to be finite. On the other hand, A. Rényi defined a \(\beta\)-expansion by means of the \(\beta\)- transformation of the unit interval \(T_ \beta x = \beta x \pmod 1\), \(x \in[0,1]\). For numbers \(x < 1\), these two expansions coincides. For \(x = 1\), the Rényi expansion is \[ d(1, \beta) = .t_ 1t_ 2 \dots, t_ k = \bigl[ \beta T^{k-1}_ \beta 1 \bigr]. \] Let \(D_ \beta\) be the set of \(\beta\)-expansions of numbers of \([0,1[\), and let \(d : [0,1] \to D_ \beta \cup \{d(1,\beta)\}\) be the function mapping \(x \neq 1\) onto its \(\beta\)-expansion, and 1 onto \(d(1,\beta)\). Let \(Fin (\beta)\) be the set of numbers \(x > 0\) having finite \(\beta\)-expansions. An algebraic integer is called a Pisot number if all its Galois conjugates have modulus less than one, and a Salem number if all its Galois conjugates are less than or equal to one in modulus and at least one conjugate has modulus one. The main results of the paper are theorems giving sufficient conditions for the inclusion \(\mathbb{Z}_ + [\beta^{-1}] \cap \mathbb{R}_ + \subset Fin (\beta)\). Theorem 2. Let \(\beta\) be the positive root of the polynomial \[ M(X) = X^ m - a_ 1X^{m-1} - a_ 2X^{m-2} - \cdots - a_ m, \quad a_ i \in \mathbb{Z},\;a_ 1 \geq a_ 2 \geq \cdots \geq a_ m > 0. \] Then \(\beta\) is a Pisot number, \(d(1, \beta) = .a_ 1a_ 2 \cdots a_ m\), and \(Fin (\beta) = (\mathbb{Z} [\beta^{-1}])_ + = \mathbb{Z} [\beta^{-1}] \cap \mathbb{R}_ +\). Theorem 3. Let \(\beta > 1\) be a real number such that \(d(1,\beta) = .t_ 1t_ 2 \dots t_ m (t_{m+1})^ \omega\), with \(t_ 1 \geq t_ 2 \geq \cdots t_ m > t_{m+1} > 0\). Then \(\beta\) is a Pisot number and \(\mathbb{Z}_ + [\beta^{-1} \subset Fin (\beta)\). Finally an application to substitution dynamical systems is given.
Reviewer: Y.Asoo (Okayama)


68W10 Parallel algorithms in computer science
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