Descriptive properties of the type of an irrational number
The type $τ$($α$) of an irrational number $α$ measures the extent to which rational numbers can closely approximate $α$. More precisely, $τ$($α$) is the infimum over those t$\in$R for which |$α$--h/k| 0. In this paper, we regard the type as a function $τ$:R\Q$\rightarrow$[1,$\infty$] and explore its descriptive properties. We show that $τ$ is invariant under the natural action of GL2(Q) on R\Q. We show that $τ$ is densely onto, and we compute the descriptive complexity of the pre-image of the singletons and of certain intervals. Finally, we show that the function $τ$ is [1,$\infty$]-upper semi-Baire class 1 complete.
math.GN↗