On beta expansions for Pisot numbers

Source abstract

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