Description of spectra of quadratic Pisot units

The spectrum of a real number β>1β>1 is the set Xm(β)Xm(β) of p(β)p(β) where p   ranges over all polynomials with coefficients restricted to A={0,1,…,m}A={0,1,…,m}. For a quadratic Pisot unit β  , we determine the values of all distances between consecutive points and their corresponding frequencies, by recasting the spectra in the frame of the cut-and-project scheme. We also show that shifting the set AA of digits so that it contains at least one negative element, or considering negative base −β instead of β, the gap sequence of the modified spectrum is a coding of an exchange of three intervals.