Comments on the spectra of Pisot numbers
For \(q\in (1,2)\) and an integer \(m\geq 1,\) write \(Y^m(q)=\{y_0=0< y_1<y_2<y_3<\dots\},\) where each \(y_i\) is expressible by a (finite) sum \(\sum_{k=0}^{\infty} u_k q^k\) with \(u_0,u_1, u_2,\dots\in \{0,\dots,m\}.\) The set \(Y^m(q)\) is called the spectra of \(q.\) The authors continue the study of the set \(Y^m(q)\) and the gap sequence \(y_1-y_0, y_2-y_1, y_3-y_2, \dots\) for Pisot numbers \(q.\) (It is known that \(q\) is a Pisot number, if and only if, \(\liminf_{k\to \infty} (y_{k+1}-y_k) >0\) for all \(m \geq 1\).) In particular, they show that if the gap \(\beta \in \{y_n-y_{n-1} : y_n \in Y^m(q)\}\}\) occurs with frequency \(\alpha\) in \(Y^m(q)\) then \(\alpha\) must be a polynomial in \(q\) with rational coefficients. Suppose \(q\) is a quadratic Pisot number, which the root of \(x^2-ax+b=0\) and \(m=[q].\) They compute the frequencies of \(1, q-a+1\) and \(a-q\) in \(Y^m(q).\)
scientific article; zbMATH DE number 5118758