Finite beta-expansions with negative bases

Abstract

International audienceThe finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $\beta$ having the negative finiteness property, that is the set of finite $(-\beta)$-expansions is equal to $\mathbb{Z}[\beta^{-1}]$. For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of $(-\beta)$-integers. We also give conditions excluding the negative finiteness property

On positional representation of integer vectors

“…and a pair of complex conjugate eigenvalues in modulus smaller than 1. As shown in[14], any element of the ring Z[β] = {a + bβ + cβ 2 : a, b, c ∈ Z} can be written as(11) a+ bβ + cβ 2 = n i=k a i β i , where k, n ∈ Z, k ≤ n and a i ∈ {0, 1} .We use the isomorphism ψ : Z[β] → Z 3 of two additive groups given by ψ(a+ bβ + cβ 2 ) = β(a + bβ + cβ 2 )) = −c + (a + c)β + (b − c)β 2 , we have ψ(β(a + bβ + cβ 2 )) =Applying this rule to Equation(11) we deduce that any element of Z 3 can be expressed as = ψ(a + bβ + cβ 2 ) = i∈I ψ(β i a i ) = n i=k T i ψ(a i ) =…”