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A certain finiteness property of Pisot number systems. (English summary) J. Number Theory 107 (2004), no. 1, 135–160. For a real number β> 1, define a map Tβ on [0, 1) by x ↦ → βx − ⌊βx⌋. For any x ∈ [0, 1), the sequence of integers xi = ⌊βT i−1 β (x) ⌋ (i = 1, 2,...) gives the greedy expansion x = ∑∞ −i i=1 xiβ of x, and, with suitable modification, leads to the β-expansion of x for any x> 0. Denoting by Fin(β) the set of x> 0 having a finite β-expansion and by Z[1/β]�0 the nonnegative elements of Z[1/β], the authors study what they call the weak finiteness property (W) of β: (W) For any x ∈ Z[1/β]�0 and any ε> 0 there exist y, z ∈ Fin(β) such that x = y − z and z < ε. They prove that if β has property (W) then it must be either a Pisot number or a Salem number. They exhibit some families of Pisot numbers having property (W). One such family is the set of cubic Pisot units. They are unable to find any Salem numbers having property (W), and it seems t

Year: 2013

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