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Entire functions mapping uncountable dense sets of reals onto each other monotonically


Author: Maxim R. Burke
Journal: Trans. Amer. Math. Soc. 361 (2009), 2871-2911
MSC (2000): Primary 03E35; Secondary 30E10
DOI: https://doi.org/10.1090/S0002-9947-09-04924-1
Published electronically: January 22, 2009
MathSciNet review: 2485411
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Abstract: When $ A$ and $ B$ are countable dense subsets of $ \mathbb{R}$, it is a well-known result of Cantor that $ A$ and $ B$ are order-isomorphic. A theorem of K.F. Barth and W.J. Schneider states that the order-isomorphism can be taken to be very smooth, in fact the restriction to $ \mathbb{R}$ of an entire function. J.E. Baumgartner showed that consistently $ 2^{\aleph_0}>\aleph_1$ and any two subsets of $ \mathbb{R}$ having $ \aleph_1$ points in every interval are order-isomorphic. However, U. Abraham, M. Rubin and S. Shelah produced a ZFC example of two such sets for which the order-isomorphism cannot be taken to be smooth. A useful variant of Baumgartner's result for second category sets was established by S. Shelah. He showed that it is consistent that $ 2^{\aleph_0}>\aleph_1$ and second category sets of cardinality $ \aleph_1$ exist while any two sets of cardinality $ \aleph_1$ which have second category intersection with every interval are order-isomorphic. In this paper, we show that the order-isomorphism in Shelah's theorem can be taken to be the restriction to $ \mathbb{R}$ of an entire function. Moreover, using an approximation theorem of L. Hoischen, we show that given a nonnegative integer $ n$, a nondecreasing surjection $ g\colon\mathbb{R}\to\mathbb{R}$ of class $ C^n$ and a positive continuous function $ \epsilon\colon\mathbb{R}\to\mathbb{R}$, we may choose the order-isomorphism $ f$ so that for all $ i=0,1,\dots,n$ and for all $ x\in\mathbb{R}$, $ \vert D^if(x)-D^ig(x)\vert<\epsilon(x)$.


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Additional Information

Maxim R. Burke
Affiliation: Department of Mathematics and Statistics, University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada C1A 4P3
Email: burke@upei.ca

DOI: https://doi.org/10.1090/S0002-9947-09-04924-1
Keywords: Order-isomorphism, second category, entire function, oracle-cc forcing, Carleman's theorem, Hoischen's theorem
Received by editor(s): March 10, 2006
Published electronically: January 22, 2009
Additional Notes: The author’s research was supported by NSERC. The author thanks F.D. Tall and the Department of Mathematics at the University of Toronto for their hospitality during the academic year 2003/2004 when much of the present paper was written.
Article copyright: © Copyright 2009 American Mathematical Society

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