Generalized radix representations and dynamical systems. IV. (English) Zbl 1190.11041
For \(\mathbf r = (r_{1},\dots , r_d)\in \mathbb R^d\) the mapping \(\tau _{\mathbf r}:\mathbb Z^d \rightarrow \mathbb Z^d\) given by
\[
\tau _{\mathbf r}(a_{1},\dots ,a_d) = (a_{2}, \dots , a_d, - \lfloor r_{1}a_{1}+\dots + r_da_d\rfloor),
\]
where \(\lfloor\cdot \rfloor\) denotes the floor function, is called a shift radix system if for each \(\mathbf a\in \mathbb Z^d\) there exists an integer \(k > 0\) with \(\tau _{\mathbf r}^k(\mathbf a) = 0\).
The authors have studied shift radix systems and their connections to \(\beta\)-expansions and canonical number systems in previous parts of this series of papers [Part I, S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, J. M. Thuswaldner, Acta Math. Hung. 108, No. 3, 207–238 (2005; Zbl 1110.11003); Part II, Acta Arith. 121, No. 1, 21–61 (2006; Zbl 1142.11055); Part III, Osaka J. Math. 45, No. 2, 347–374 (2008; Zbl 1217.11007)].
The aim of the current paper is to establish the distribution of Pisot polynomials with and without the finiteness property (F).
Let \({\mathcal D_{d-1}}\) and \(\mathcal D_{d-1}^0\) be the sets of vectors \(\mathbf r\) in \({\mathbb R}^{d-1}\) such that for all \({\mathbf a}\in{\mathbb Z}^{d-1}\), the sequence \((\tau^k_{{\mathbf r}}({\mathbf a}))_{k\geq 0}\) is ultimately periodic resp. reaches \(0\).
The sets \({\mathcal B}_d(M)\) and \({\mathcal B}_d(M)^0\) are defined to be the sets of vectors \((b_2,\ldots, b_d)\in{\mathbb Z}^{d-1}\) such that \(X^d-MX^{d-1}-b_2X^{d-2}-\cdots-b_d\) is a Pisot or Salem polynomial resp. a Pisot polynomial with property (F). Here, as usual, a Pisot polynomial is a monic irreducible polynomial over \(\mathbb Z\) which has one real root greater than one and all other roots are located in the unit disk. A Salem polynomial is a monic irreducible polynomial over \(\mathbb Z\) which has one real root greater than one and all other roots are located in the closed unit disk and at least one of them has modulus \(1\). A Pisot polynomial is said to have property (F), if every non-negative element of \({\mathbb Z}[1/\beta]\) has a finite greedy \(\beta\)-expansion, where \(\beta\) denotes the dominant root of the polynomial.
The main results of the paper are that for \(d\geq 2\), \[ \left| \frac{|{\mathcal B}_d(M)|}{M^{d-1}}-\lambda_{d-1}({\mathcal D}_{d-1})\right|=O(M^{-1/(d-1)}) \] and \[ \lim_{M\to\infty}\frac{|{\mathcal B}_d^0(M)|}{M^{d-1}}=\lambda_{d-1}({\mathcal D}_{d-1}^0), \] where \(\lambda_{d-1}\) denotes the \((d-1)\)-dimensional Lebesgue measure.
The authors have studied shift radix systems and their connections to \(\beta\)-expansions and canonical number systems in previous parts of this series of papers [Part I, S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, J. M. Thuswaldner, Acta Math. Hung. 108, No. 3, 207–238 (2005; Zbl 1110.11003); Part II, Acta Arith. 121, No. 1, 21–61 (2006; Zbl 1142.11055); Part III, Osaka J. Math. 45, No. 2, 347–374 (2008; Zbl 1217.11007)].
The aim of the current paper is to establish the distribution of Pisot polynomials with and without the finiteness property (F).
Let \({\mathcal D_{d-1}}\) and \(\mathcal D_{d-1}^0\) be the sets of vectors \(\mathbf r\) in \({\mathbb R}^{d-1}\) such that for all \({\mathbf a}\in{\mathbb Z}^{d-1}\), the sequence \((\tau^k_{{\mathbf r}}({\mathbf a}))_{k\geq 0}\) is ultimately periodic resp. reaches \(0\).
The sets \({\mathcal B}_d(M)\) and \({\mathcal B}_d(M)^0\) are defined to be the sets of vectors \((b_2,\ldots, b_d)\in{\mathbb Z}^{d-1}\) such that \(X^d-MX^{d-1}-b_2X^{d-2}-\cdots-b_d\) is a Pisot or Salem polynomial resp. a Pisot polynomial with property (F). Here, as usual, a Pisot polynomial is a monic irreducible polynomial over \(\mathbb Z\) which has one real root greater than one and all other roots are located in the unit disk. A Salem polynomial is a monic irreducible polynomial over \(\mathbb Z\) which has one real root greater than one and all other roots are located in the closed unit disk and at least one of them has modulus \(1\). A Pisot polynomial is said to have property (F), if every non-negative element of \({\mathbb Z}[1/\beta]\) has a finite greedy \(\beta\)-expansion, where \(\beta\) denotes the dominant root of the polynomial.
The main results of the paper are that for \(d\geq 2\), \[ \left| \frac{|{\mathcal B}_d(M)|}{M^{d-1}}-\lambda_{d-1}({\mathcal D}_{d-1})\right|=O(M^{-1/(d-1)}) \] and \[ \lim_{M\to\infty}\frac{|{\mathcal B}_d^0(M)|}{M^{d-1}}=\lambda_{d-1}({\mathcal D}_{d-1}^0), \] where \(\lambda_{d-1}\) denotes the \((d-1)\)-dimensional Lebesgue measure.
Reviewer: Clemens Heuberger (Graz)
MSC:
| 11K16 | Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. |
| 11A63 | Radix representation; digital problems |
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 37B10 | Symbolic dynamics |
Keywords:
beta expansion; canonical number system; periodic point; contracting polynomial; Pisot numberReferences:
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