| Pisot numbers have a long history, being studied as early as 1912 by Thue [38]. Some simple examples of Pisot numbers are the golden ratio (approximately 1.6180339), all integers greater than or equal to 2, and the real root of x3 − x − 1 (approximately 1.324717957). Salem shows that the set of Pisot numbers is infinite and very structured [31]. Pisot numbers appear in a variety of different areas of mathematical research, such as: β-expansions [24], disjoint coverings of the natural numbers [2], robotics [11], quasilattices and quasicrystals [9], exceptional sets in harmonic analysis [31], and Salem numbers [31, 32]. A recent line of inquiry, initiated by Erdös, Joó and Komornik [13], is the determination of l1(q) for Pisot numbers q. Here lm( q) := inf{|y| : y = ε nqn + εn −1qn −1 ε+…+ε0, ε i ∈ {±m, ±(m − 1),…,±1, 0}, y ≠ 0}. This line of inquiry is generalized by considering the spectra of any set of polynomials evaluated at q, where the coefficients of the polynomials are restricted to a finite set of integers. Some common generalizations include restricting the coefficients to {0,1} or {±1}.;An algorithm for computing lm( q) and its generalizations is given in this thesis. Using this algorithm a systematic investigation of the spectra of {±1} polynomials is done. This investigation results in the discovery of non-Pisot numbers with discrete spectra. Furthermore a complete description of lm( q) for all unit quadratic Pisot numbers is given. A similar description appears to be possible for some cubic Pisot numbers, but so far no proof is known. This class of cubic Pisot numbers is studied. |
