Periods of β-expansions and linear recurrent sequences

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.