The arithmetic and geometry of Salem numbers

Small Salem Numbers

Salem numbers of negative trace

We prove that, for all d ≥ 4, there are Salem numbers of degree 2d and trace −1, and that the number of such Salem numbers is d/ (log log d). As a consequence, it follows that the number of totally positive algebraic integers of degree d and trace 2d − 1 is also d/ (log log d).

Salem Numbers, Pisot Numbers, Mahler Measure, and Graphs

We use graphs to define sets of Salem and Pisot numbers, and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers n the smallest known element of the n-th derived set of the set of Pisot numbers comes from a graph. We define the Mahler measur...

Some computations on the spectra of Pisot and Salem numbers

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdős, Joó and Komornik in 1990, is the determination of l(q) for Pisot numbers q, where l(q) = inf(|y| : y = 0 + 1q + · · ·+ nq, i ∈ {±1, 0}, y 6= 0). Although the quantity l(q) is known for some Pisot numbers q, there has been no general method for computing l(q). This paper gives such an algorithm. ...

There Are Salem Numbers of Every Trace

We show that there are Salem numbers of every trace. The nontrivial part of this result is for Salem numbers of negative trace. The proof has two main ingredients. The first is a novel construction, using pairs of polynomials whose zeros interlace on the unit circle, of polynomials of specified negative trace having one factor a Salem polynomial, with any other factors being cyclotomic. The sec...