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Topic: a question on real-analytic functions
Replies: 9   Last Post: Mar 16, 2002 10:05 AM

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Dave L. Renfro

Posts: 4,546
Registered: 12/3/04
Re: a question on real-analytic functions
Posted: Mar 15, 2002 12:39 PM
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David Bernier <ezcos@yahoo.com>
[sci.math 13 Mar 2002 23:43:19 -0800]
http://mathforum.org/epigone/sci.math/zumrizom

wrote

> Suppose f: ]0,1[ -> Reals is analytic at every point in
> ]0, 1[ and that f(r) is rational for every rational r in
> the domain of definition.
>
> Some examples: x|-> 1/x
> x|-> P(x), P in Q[x],
> x|-> 9/23* (1/(x^2 + 1))
>
> The most general form I can come up with is:
>
> - P in Q[x],
>
> - T in Q[x], such that T has no zeros inside ]0, 1[
>
> - Let f(x) = P(x)/T(x) .
>
> That's all I could think of. I chose the domain of definition
> ]0, 1[ to be small, for example compared to the Reals .
>
> So I'm curious to know whether any other such functions
> f exist.
>
> BTW: The set of all such f forms a Q-algebra, it seems.


In this same thread Robert Israel mentioned an earlier post of his:

http://mathforum.org/epigone/sci.math/khoustelwhand/6fkids$cjq$1@nntp.ucs.ubc.ca
http://groups.google.com/groups?selm=6fkids%24cjq%241%40nntp.ucs.ubc.ca

Israel showed that there are c many analytic functions that
map the rationals to the rationals and then points out that there
are only countably many functions of the type you've described.

In case anyone has a further interest in this, I thought I
mention a few papers related to this question.

[1] Philip Franklin, "Analytic transformations of everywhere dense
point sets", Trans. Amer. Math. Soc. 27 (1925), 91-100.
[Announced in Bull. Amer. Math. Soc. 30 (1924), 482.]
[JFM 51.0166.01]

Theorem I (p. 94): "For any two enumerable linear point sets,
each everywhere dense on an open interval,
an analytic function can be found which maps
the two intervals on one another, and effects
a one to one correspondence between the point
sets."

[[ NOTE: I don't believe that any of the following later references
mention Franklin's paper ]]

[2] Paul Erdos, "Some unsolved problems", Michigan Math. J. 4 (1957),
291-300. [MR 20 #5157; Zbl 81.00102]

Problem #24 (p. 197): "Does there exist an entire function f, not
of the form f(x) = a + bz, such that the
number f(x) is rational or irrational
according to as x is rational or irrational?
More generally, if A and B are two
denumerable, dense sets, does there exist
an entire function which maps A onto B?"

[3] Fred Gross, "Research Problem #19", Bull. Amer. Math. Soc. 71
(1965), 853.

[[ Gross apparently asks essentially the same question, but I
have not actually looked at this reference. Maurer claims
that his result answers negatively the question Gross asked,
but see Gross' comment at the end of the Zbl review of
Maurer's paper (URL given in [4] below). ]]

[4] Ward D. Maurer, "Conformal equivalence of countable dense sets",
Proc. Amer. Math. Soc. 18 (1967), 269-270.
[MR 35 #6829; Zbl 189.36204]
http://www.emis.de/cgi-bin/Zarchive?an=0189.36204

Theorem: "Let A and B be two countable dense sets in the complex
plane. Then there exists an entire function taking A
onto B."

[5] Karl F. Barth and Walter J. Schneider, "Entire functions
mapping countable dense subsets of the reals onto each other
monotonically", J. London Math. Soc. (2) 2 (1970), 620-626.
[MR 42 #4729; Zbl 201.09203]
http://www.emis.de/cgi-bin/Zarchive?an=0201.09203

MR (by W. D. Maurer): "This paper provides further information
about a problem due to Erdos concerning the
existence of functions mapping arbitrary
countable dense sets onto others. The reviewer
[Proc. Amer. Math. Soc. 18 (1967), 269-270;
MR 35 #6829] showed that, given any two
countable dense subsets of the complex plane,
there exists an entire function taking one of
these onto the other. In this paper we are
talking about the line, not the plane, but
the conclusion is stronger: the function takes
points of the first countable dense set, and
only such points, into points of the second
one. The argument is very tedious and, in fact,
is not completely carried out; only the case
in which both of the given sets are the
rationals is completed (the function involved
in this case, however, is transcendental, as
it is in the general case). As an application,
the authors prove that strictly monotone
increasing entire functions may assume any
sequence of values whatsoever at all positive
integers, just so long as these values
themselves form an increasing sequence."

[6] Stuart Rankin and Daihachiro Sato, "Entire functions mapping
countable dense subsets of the reals onto each other
monotonically", Bull. Aust. Math. Soc. 10 (1974), 67-70.
[MR 49 #10883; Zbl 275.30020]

MR (by T. Kovari): "By modifying a technique of W. D. Maurer,
the authors give a new proof of the following
result of K. F. Barth and W. J. Schneider: Let
A and B be countable dense subsets of the real
line; then there exists a transcendental entire
function f such that the restriction of f to the
real line is a real homeomorphism, and f(A) = B."

[7] Emil A. Corena, "Problem #10361", Amer. Math. Monthly, 101
(1994), 175.

"Do there exist nonlinear C^1 functions f:R --> R such
that for any rational x, f(x) is also rational and for
any irrational x, f(x) is also irrational?"


Dave L. Renfro







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