Some computations on the spectra of Pisot and Salem numbers (Q2781224)

For any real \(q \in (1,2)\) the sets composed by the numbers \(\varepsilon_0+\varepsilon_1 q+\dots+\varepsilon_n q^{n},\) where \(\varepsilon_i\) belong to a finite set of integers \(S,\) were studied earlier by Erdős, Joó and Joó, Komornik, Loreti, Bugeaud, Peres, Solomyak and others. Here, these sets are called the spectrum of \(q\) with respect to \(S.\) Let \(\Lambda(q)\) and \(A(q)\) be these sets corresponding to \(S=\{0,1,-1\}\) and \(S=\{-1,1\},\) respectively, so that \(A(q) \subset \Lambda(q).\) One of the problems important in this area is to determine the smallest non-zero element in \(\Lambda(q)\) for a Pisot number \(q.\) (It is not known whether there are non-Pisot numbers \(q\) for which each non-zero element of \(\Lambda(q)\) is bounded away from zero.)NEWLINENEWLINEThe authors give an algorithm for this and calculate this smallest element for Pisot numbers up to degree \(9.\) They also give a table of Pisot numbers of degree up to seven when this smallest element coincides with the smallest non-zero element in the set \(A(q).\) They also answer a question of Peres and Solomyak and show that there are non-Pisot numbers \(q\) for which the set \(A(q)\) is not dense. More precisely, it is shown that if \(q>1\) is the root of the polynomial \(x^n-x^{n-1}-\dots-x+1,\) \(n \geq 4,\) then the spectrum \(A(q)\) is discrete, namely, for any finite interval \([a,b]\) of the real line, the set \(A(q) \cap [a,b]\) has only a finite number of elements. (These are all Salem numbers. The smallest such number found is \(1.72208 \dots\). It corresponds to \(n=4\).)NEWLINENEWLINESeveral related questions are also considered. For instance, the authors determine \(16\) Pisot polynomials (all of degree from \(6\) to \(10\)) that do not divide a \(\pm 1\) polynomial. For example, \(x^6-x^5-2x^4+x^2-x-1\) is one of these. Of course, \(q\) is such a Pisot number (a root of such a Pisot polynomial) precisely when \(0 \in A(q).\)