Comments on the spectra of Pisot numbers Original Research Article

For qset membership, variant(1,2), Erdős, Joó and Komornik studied the spectra of q, defined asimage Feng and Wen showed that for q a Pisot number, the gap sequencey1−y0,y2−y1,…,yk+1−yk,… is the iterative fixed point of a substitution. The second author used this substitution to determine the frequency of particular gap sizes in the spectra, and gave a detailed account when q is the golden ratio. In this paper we give some remarkable properties for this substitution, and the incidence matrix associated with it. In particular, if P(x) is the characteristic polynomial of the incidence matrix, and p(x) the minimal polynomial of the Pisot number q, then p(x)P(x). Moreover, q is an eigenvalue of maximal modulus. As a corollary of this, an open question of the second authorʹs regarding the frequencies of gap sizes is answered. We also give conditions under which the gap frequencies are guaranteed to exist. In addition, we show that P(x) can be used to describe the index ik where yik=qk in Ym(q). Lastly, substitutions and frequencies are determined precisely for two classes of Pisot numbers.