Some comments on Garsia numbers

AGarsia number is an algebraic integer of norm ±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 21/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 > √ 2. • There exist two isolated limit points of the positive real Garsia numbers greater than √ 2. These are 1.618 · · · and 1.465 · · · , the roots of z2−z−1 and z3 − z2 − 1, respectively. There are no other limit points greater than √ 2. • There exist infinitely many limit points of the positive real Garsia numbers, including λm,n, the positive real root of zm − zn − 1, with m > n.