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

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 $x\in \mathbb {R}$, $|D^if(x)-D^ig(x)|<\epsilon (x)$.