Periods of β-expansions and linear recurrent sequences
Yan-Hui Qu, Hui Rao, Ya-min Yang · Acta Arithmetica · 2005
Abstract. Let β> 1 be a Pisot number. It is well known that a number x has periodic β-expansion if and only if x ∈ Q(β).When β is a quadratic Pisot unit, we show that the period of the β-expansion of x is determined by a linear recurrent sequence related to β and x. Particularly, if β = ( 5+1)/2 is the golden number, then the periods of the β-expansions are determined by the prominent Fibonacci sequence. 1.