On beta expansions for Pisot numbers

Abstract

Abstract. Given a number β > 1, the beta-transformation T = T β is defined for x ∈ [0, 1] by T x := βx (mod 1). The number β is said to be a betanumber if the orbit {T n (1)} is finite, hence eventually periodic. In this case β is the root of a monic polynomial R(x) with integer coefficients called the characteristic polynomial of β. If P(x) is the minimal polynomial of β, then R(x) = P(x)Q(x) for some polynomial Q(x). It is the factor Q(x) which concerns us here in case β is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether Q(x) must be cyclotomic in this case, particularly if 1 < β < 2. We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in [1, 1.9324] ∪ [1.9333, 1.96] (an infinite set), by a search up to degree 50 in [1.9, 2], to degree 60 in [1.96, 2], and to degree 20 in [2, 2.2]. We find the smallest counterexample, the counterexample of smallest degree, examples where Q(x) is nonreciprocal, and examples where Q(x) is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to 2 from above, and infinite sequences of β with Q(x) nonreciprocal which converge to 2 from below and to the 6th smallest limit point of the Pisot numbers from both sides. We conjecture that these are the only limit points of such numbers in [1,2]. The Pisot numbers for which Q(x) is cyclotomic are related to an interesting closed set of numbers F introduced by Flatto, Lagarias and Poonen in connection with the zeta function of T . Our examples show that the set S of Pisot numbers is not a subset of F.

Minimal polynomials of some beta-numbers and Chebyshev polynomials

“…While our main interest in this paper is irreducible beta-polynomials, Boyd [2] focused on reducible beta-polynomials. Specifically, he dealt with the situation where β is a Pisot number.…”

On univoque Pisot numbers

“…These sequences of Pisot numbers are the regular Pisot numbers associated with ψ r . See for example [6,14].…”

“…Then for sufficiently large n, the polynomials P χ (x)x n ± A χ (x) and P χ (x)x n ± B χ (x) admit a unique root between 1 and 2, which is a Pisot number. See for example [6,14].…”

There are Only Eleven Special Pisot Numbers

“…This can be done using the Dufresnoy-Pisot algorithm, as developed by Boyd [3]. In fact, Boyd himself carried out such a search up to degree 30 (private communication), finding no more special Pisot numbers.…”

“…The original algorithm was developed for finding all Pisot numbers in an interval containing no limit points of Pisot numbers. It was readily adapted to finding all Pisot numbers of bounded degree in any interval (as remarked in [3]). This completes the proof of Theorem 1.…”