The following example gives a connection between descriptive set theory and the theory of approximation by algebraic numbers:

There exists a  classification, due to Mahler, of real (and complex) numbers into four classes $A, S, T$ and $U$ according
to their properties of approximation by algebraic numbers. 

In the paper [The Borel Classes of Mahler's $A$, $S$, $T$, and $U$ Numbers](http://www.jstor.org/stable/2160682?seq=1#page_scan_tab_contents), the author studies these classes from the point of view of Descriptive Set Theory, and determines their complexity in the Borel hierarchy.