On spectra of neither Pisot nor Salem algebraic integers (Q2655195)

Let \(q>1\) be a real number, and let \(A(q)\) (resp. \(\Lambda(q)\)) be the set of values of polynomials with coefficients \(\{-1,1\}\) (resp. \(\{-1,0,1\}\)) at \(q\). A spectrum \(A(q)\) (resp. \(\Lambda(q)\)) of \(q\) is called discrete if the set \(A(q)\) (resp. \(\Lambda(q)\)) has only finitely many elements in any finite interval \([a,b]\). Of course, the spectrum can only be discrete if \(q\) is an algebraic number. In particular, both \(A(q)\) and \(\Lambda(q)\) are discrete for every Pisot number \(q>1\), but there are also other algebraic numbers with this property for \(A(q)\). (The statement that \(\Lambda(q)\) is discrete if and only if \(q>1\) is a Pisot number is a major open problem in this area.) In this paper, the author proves that if \(1<q<2\) and \(A(q)\) is discrete then all the real conjugates of \(q\) are of modulus strictly smaller that \(q\). By an open conjecture of Borwein and Hare, such \(q\) must be a Perron number, i.e., all (real and complex) conjugates of \(q\) must be of moduli smaller than \(q\), so the paper settles a part of the conjecture concerning the real conjugates.