Some comments on Garsia numbers

Abstract:

A Garsia number is an algebraic integer of norm $\pm 2$ such that all of the roots of its minimal polynomial are strictly greater than $1$ in absolute value. Little is known about the structure of the set of Garsia numbers. The only known limit point of positive real Garsia numbers was $1$ (given, for example, by the set of Garsia numbers $2^{1/n}$). Despite this, there was no known interval of [1,2] where the set of positive real Garsia numbers was known to be discrete and finite. The main results of this paper are:

An algorithm to find all (complex and real) Garsia numbers up to some fixed degree. This was performed up to degree $40$.

An algorithm to find all positive real Garsia numbers in an interval $[c, d]$ with $c > \sqrt {2}$.

There exist two isolated limit points of the positive real Garsia numbers greater than $\sqrt {2}$. These are $1.618\cdots$ and $1.465\cdots$, the roots of $z^2-z-1$ and $z^3-z^2-1$, respectively. There are no other limit points greater than $\sqrt {2}$.

There exist infinitely many limit points of the positive real Garsia numbers, including $\lambda _{m,n}$, the positive real root of $z^m -z^n - 1$, with $m > n$.