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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Entire functions mapping uncountable dense sets of reals onto each other monotonically
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by Maxim R. Burke PDF
Trans. Amer. Math. Soc. 361 (2009), 2871-2911 Request permission

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 $x\in \mathbb {R}$, $|D^if(x)-D^ig(x)|<\epsilon (x)$.
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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
  • 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.
  • © Copyright 2009 American Mathematical Society
  • 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
  • MathSciNet review: 2485411