From: renfrod@central.edu (Dave L. Renfro)
Subject: Re: Subset of the square
Date: 24 Nov 2000 10:35:52 -0500
Newsgroups: sci.math
Summary: [missing]

Fred Galvin <galvin@math.ukans.edu>
[sci.math Thu, 23 Nov 2000 22:31:17 -0600]
<http://forum.swarthmore.edu/epigone/sci.math/koxjalnil>

wrote

> On 23 Nov 2000, Dave L. Renfro wrote:
> 
>> Joseph levine <jolev@hotmail.com>
>> [sci.math 23 Nov 00 02:29:16 -0500 (EST)]
>> <http://forum.swarthmore.edu/epigone/sci.math/koxjalnil>
>> 
>> wrote
>> 
>> > Hello:
>> > 
>> > Does there exist a subset of the unit square such that all
>> > horizontal lines intersect the set in countably many points but
>> > all vertical lines intersect the set in uncountably many points?
>> 
>> See the sci.math thread "measure"
>> [Claude Danthony; Dec. 3, 1996] at
>> 
>>
<http://forum.swarthmore.edu/epigone/sci.math/junfralcland/32A4016F.59D3@umpa.ens-lyon.fr>
>> 
>> To show you how to fish, all I did was go to the Math Forum
>> sci.math search web page at
>> 
>> <http://forum.swarthmore.edu/discussions/epi-search/sci.math.html>
>> 
>> and enter certain key words. My first attempt was 'horizontal',
>> 'vertical', and 'countable', and this gave me the URL above.
> 
> OK, I tried that, and sure enough, I found an old sci.math thread
> containing the keywords 'horizontal', 'vertical', and 'countable'.
> Does one of the messages in that thread answer Joseph levine's
> question? Mostly it seems to be about more complicated stuff.

I gave the URL, Claude Danthony, above (a reply to David Madore's
reply to pcascini@hotmail.com's Nov. 30, 1996 thread). I actually
got 6 or 7 threads, with maybe 12 to 15 URL's total. Most of the
3 or 4 minutes I spent was in waiting for the web pages to appear,
as only few seconds were enough to see that (with one exception)
they were not appropriate. In part, Danthony writes

% A classical counterexemple to Fubini theorem (see Rudin) 
% is the following (due to sierpinski):
% 
% Using the continuum hypothesis, one can give R an order such
% that for each x, {y<x} is countable. Consider the set A in the
% closed square  A={ (x,y) ; y<x }.
% This set as the strange property that each horizontal cut is 
% countable, and each vertical cut has a countable complement.

I didn't notice this yesterday, but now I notice that the example
assumes CH, which I guess is cheating (unless the result itself is 
independent of ZFC, which it isn't). But you do get "countable
complement", which is a heck of a lot more than just "uncountable",
CH or no CH.

I like your example of the 2'nd coordinate map of any square-filling
parametric curve (given in another post in this thread). Basically,
what you're getting is a continuous function all of whose nonempty
level sets are perfect sets. Such functions (when they fail to be
locally constant anywhere) are often called "locally recurrent" in
the literature. Nina Bary, on p. 640 of [1] was, I think, the
first to observe the existence of such a function, with Singh [6]
(1931) and Gillis [3] (1939) independently discovering them. Bush [2]
gives an arithmetical construction of such a function and introduced
the term "locally recurrent" (Bush was unaware of the Bary, Singh,
and Gillis papers, and so Bush independently formulated the property
and his example) and Marcus (aware of the Gillis paper, but not
Bary's or Singh's papers) discusses several issues related to
locally recurrent functions. There are, of course, many other
papers on this topic, but I've probably strayed too far from the
original thread topic as it is! [Perhaps I should mention that 
the locally recurrent functions only form a first category subset
of C[0,1] (sup norm), and so their existence cannot be obtained
by using a Baire category argument in this space. (This doesn't
prevent their existence from being obtained by using a topologically
different norm, but I'm not aware of anyone having done this.)]

[1] Nina Bary, "Memoire sur la representation finie des fonctions
    continues", Math. Ann. 103 (1930), 185-248, 598-653.

[2] K. A. Bush, "Locally recurrent functions", Amer. Math. Monthly
    69 (1962), 199-206.

[3] J. Gillis, "Note on a conjecture of Erdos", Quarterly J. Math.
    (Oxford), 10 (1939), 151-154.

[4] Solomon Marcus, "On locally recurrent functions", Amer. Math.
    Monthly 70 (1963), 822-826.

[5] A. Singh, "On the method of construction and some properties
    of a class of non-differentiable functions", Proc. Benares
    Math. Soc. 13 (1931), 1-17.

Dave L. Renfro