Salem numbers and Pisot numbers via interlacing

TL;DR

This paper introduces a unified interlacing construction method for Salem and Pisot numbers, demonstrating that all Salem numbers can be generated this way and their limit points are Pisot numbers, supporting Boyd's conjecture.

Contribution

It provides a general construction linking Salem and Pisot numbers through interlacing rational functions, unifying previous results and confirming Boyd's conjecture.

Findings

All Salem numbers can be obtained via the interlacing construction.

Limit points of Salem numbers in this construction are Pisot numbers.

The construction encompasses all Pisot numbers as well.

Abstract

We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the "obvious" limit points of the set of Salem numbers produced by our theorems, and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we show that all Salem numbers are produced via an interlacing construction.