The Borel classes of Mahler’s $A$, $S$, $T$, and $U$ numbers

Abstract:

In this article we examine the A, S, T, and U sets of Mahler’s classification from a descriptive set theoretic point of view. We calculate the possible locations of these sets in the Borel hierarchy. A turns out to be $\Sigma _2^0$-complete, while U provides a rare example of a natural $\Sigma _3^0$-complete set. We produce an upperbound of $\Sigma _4^0$ for S and show that T is $\Pi _4^0$ but not $\Sigma _3^0$. Our main result is based on a deep theorem of Schmidt that allows us to guarantee the existence of the T numbers.