Littlewood Pisot numbers (Q2490149)

The author studies Littlewood Pisot numbers which are defined as Pisot numbers whose minimal polynomials are Littlewood polynomials, i.e., those with \(\pm 1\) coefficients. He also defines Littlewood Salem numbers as Salem numbers which are roots of (not necessarily irreducible) Littlewood polynomials. The main theorem of this paper asserts that every Littlewood Pisot number has the minimal polynomial \(x^n-x^{n-1}-x^{n-2}-\dots-x-1=0,\) where \(n \geq 2\). The proof involves Boyd's algorithm for determining all Pisot numbers in a given interval. The author also proves that every Littlewood Pisot number is a limit point of Littlewood Salem numbers from below. The proof uses some arguments of Borwein, Dobrowolski and Mossinghoff. They earlier proved this for the golden mean \((1+\sqrt{5})/2\) which is the smallest Littlewood Pisot number. Finally, he proves a result which implies that every reciprocal Littlewood polynomial of odd degree has at least three unimodular roots.