Descriptive properties of the type of an irrational number

TL;DR

This paper investigates the descriptive and topological properties of the function assigning to each irrational number its approximation type, revealing its invariance, density, and complexity characteristics.

Contribution

It introduces the function τ mapping irrationals to their approximation types and analyzes its invariance, density, and descriptive complexity properties.

Findings

τ is invariant under GL2(Q) action.

τ is densely onto.

τ is upper semi-Baire class 1 complete.

Abstract

The type $\tau$($\alpha$) of an irrational number $\alpha$ measures the extent to which rational numbers can closely approximate $\alpha$. More precisely, $\tau$($\alpha$) is the infimum over those t$\in$R for which |$\alpha$--h/k|<k^{--t--1} has at most finitely many solutions h,k$\in$Z, k>0. In this paper, we regard the type as a function $\tau$:R\Q$\rightarrow$[1,$\infty$] and explore its descriptive properties. We show that $\tau$ is invariant under the natural action of GL2(Q) on R\Q. We show that $\tau$ 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 $\tau$ is [1,$\infty$]-upper semi-Baire class 1 complete.