From: dlrenfro@gateway.net (Dave L. Renfro)
Subject: Re: Question from Rudin
Date: 11 May 2000 10:36:18 -0400
Newsgroups: sci.math
Summary: [missing]

paul mitchell <paul1@cats.ucsc.edu>
[sci.math Wed, 10 May 2000 12:33:09 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/flarzhozal>

wrote

> This question from Rudin's textbook, p44 
> has been stumping the undergrad Real Analysis
> class at UCSC:  Is there a non-empty perfect
> set in R that contains no rational?  We've tried
> various translations of the Cantor set, but haven't
> been able to demonstrate both closedness and the
> absence of rationals.  Any hints?

Any translation of the Cantor set is closed. There are many
ways to prove this. One way is to observe that the Cantor set
is compact, any translation is a continuous map, any continuous
image of a compact set is compact, and any compact set is
closed. [Or: The Cantor set is closed, any translation is a
homeomorphism, and any homeomorphic image of a closed set
is closed.]

In 1884, Scheeffer [4] (pp. 291-293) published a proof of
the following:

Let C be a perfect nowhere dense set of reals and Z be a
countable set of reals. Then for each pair of real numbers
a1 < a2, there exists a real number b such that a1 < b < a2
and the b-translate of C has empty intersection with Z.

For a proof see pp. 52-53 of Young/Young [5] (these page numbers
are for the Chelsea edition). Young/Young's proof actually shows
that the set of real numbers b for which the b-translate of
C contains no points of Z is the complement of a first category
set of real numbers. In particular, there are continuum many such
b's in every open interval.

Boes/Darst/Erdos prove in [2] that given any countable set Z, the
set of b's in (0,1] such that the symmetric Cantor set, formed in
the following manner, is disjoint from Z is the complement of a
first category subset of (0,1]: Remove from [0,1] a segment of 
length b/3 so as to leave two intervals of equal length; from
each of these two intervals remove a segment of length b/9 to
leave 4 intervals of equal length; continue in this manner.

In 1959, Bagemihl proved the following stronger version of
Scheeffer's theorem. [See also Morgan [3], pp. 194-195.]

Let F be a first category set of reals and Z be a countable set
of reals. Then the set of real numbers b for which the b-translate
of F contains no points of Z is the complement of a first category
set of real numbers.

[1] Frederick Bagemihl, "A note on Scheeffer's theorem", Michigan
    Math. J. 2 (1954), 149-150.

[2] Duane Boes, Richard Darst, and Paul Erdos, "Fat, Symmetric,
    Irrational Cantor Sets", The American Mathematical Monthly
    88 (1981), 340-341.

[3] John C. Morgan II, POINT SET THEORY, Pure and Applied 
    Mathematics 131, Marcel Dekker, 1989. [QA 603 .P67]

[4] L. Scheeffer, "Zur Theorie der stetigen Functionen einer
    reellen Veränderlichen", Acta Math. 5 (1884), 279-296.
    [JFM 16.0340.01 at <http://www.emis.de/MATH/JFM/JFM.html>]

[5] Grace C. Young and William H. Young, THE THEORY OF SETS OF
    POINTS, Cambridge, 1906. [2'nd edition (actually, just a
    revised reprint with an appendix of additional notes) was
    published by Chelsea in 1972, and can be found in most college
    libraries in the QA 248's.]

Dave L. Renfro