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The Borel classes of Mahler's $ A$, $ S$, $ T$, and $ U$ numbers


Author: Haseo Ki
Journal: Proc. Amer. Math. Soc. 123 (1995), 3197-3204
MSC: Primary 04A15; Secondary 11J81
DOI: https://doi.org/10.1090/S0002-9939-1995-1273503-6
MathSciNet review: 1273503
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Abstract: In this article we examine the A, S, T, and U sets of Mahler's classification from a descriptive set theoretic point of view. We calculate the possible locations of these sets in the Borel hierarchy. A turns out to be $ \Sigma _2^0$-complete, while U provides a rare example of a natural $ \Sigma _3^0$-complete set. We produce an upperbound of $ \Sigma _4^0$ for S and show that T is $ \Pi _4^0$ but not $ \Sigma _3^0$. Our main result is based on a deep theorem of Schmidt that allows us to guarantee the existence of the T numbers.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1273503-6
Keywords: Borel hierarchy, completeness, descriptive set theory, hardness, Mahler's classification
Article copyright: © Copyright 1995 American Mathematical Society

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