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, the author studies these classes from the point of view of Descriptive Set Theory, and determines their complexity in the Borel hierarchy.
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