Some computations on the spectra of Pisot and Salem numbers
Peter Borwein, Kevin G. Hare · Mathematics of Computation · 2001
Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdős, Joó and Komornik in 1990, is the determination of l ( q ) l(q) for Pisot numbers q q , where \[ l ( q ) = inf ( | y | : y = ϵ 0 + ϵ 1 q 1 + ⋯ + ϵ n q n , ϵ i ∈ { ± 1 , 0 } , y ≠ 0 ) . l(q) = \inf (|y|: y = \epsilon _0 + \epsilon _1 q^1 + \cdots + \epsilon _n q^n, \epsilon _i \in \{\pm 1, 0\}, y eq 0). \] Although the quantity l ( q ) l(q) is known for some Pisot numbers q q , there has been no general method for computing l ( q ) l(q) . This paper gives such an algorithm. With this algorithm, some properties of l ( q ) l(q) and its generalizations are investigated. A related question concerns the analogy of l ( q ) l(q) , denoted