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- W4384264721 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 $tau$:RQ$rightarrow$[1,$infty$] and explore its descriptive properties. We show that $tau$ is invariant under the natural action of GL2(Q) on RQ. 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." @default.
- W4384264721 created "2023-07-14" @default.
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- W4384264721 date "2023-07-12" @default.
- W4384264721 modified "2023-10-17" @default.
- W4384264721 title "Descriptive properties of the type of an irrational number" @default.
- W4384264721 doi "https://doi.org/10.48550/arxiv.2307.05965" @default.
- W4384264721 hasPublicationYear "2023" @default.
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