Infinite families of totally positive integers with small trace

The absolute trace of a totally positive algebraic integer is defined to be the average of its conjugate elements in $\mathbb{R}$. We show that there are infinitely many totally positive algebraic integers of absolute trace at most $1.81$ but only finitely many of absolute trace at most $1.80$. Previously, it was only known that there are infinitely many totally positive algebraic integers of absolute trace at most $2$. The infinite family of totally positive integers with small trace is constructed non-explicitly via a combination of geometry of numbers and potential theory. This talk is of work in progress.