DESCRIPTIVE PROPERTIES OF THE TYPE OF AN IRRATIONAL NUMBER
Résumé
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∈R for which |α−h/k|0. In this paper, we regard the type as a function τ:R\Q→[1,∞] 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,∞]-upper semi-Baire class 1 complete.
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