Description of spectra of quadratic Pisot units

The spectrum of a real number $\beta >1$ is the set $X^m(\beta )$ of $p(\beta )$ where $p$ ranges over all polynomials with coefficients restricted to $\mathcal{A}=\{0,1,\dots ,m\}$. For a quadratic Pisot unit $\beta$, 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 $\mathcal{A}$ of digits so that it contains at least one negative element, or considering negative base $-\beta$ instead of $\beta$, the gap sequence of the generalized spectrum is a coding of an exchange of three intervals.