Does there
exist a Baire category version of a Nikodym set?
by
Dave Renfro
(RENFRO@ALPHA.NSULA.EDU)
Recall that a Nikodym set (in the unit square of
) is a set
of full planar measure having the property that for each of its
points there exists a line through that point having only that point
as its intersection with the set. The name comes from O. Nikodym,
who published a construction of such a set in 1927. See p. 100 of
Falconer's 1985 book "The Geometry of Fractal Sets". In 1952,
R. O. Davies showed there exists a Nikodym set having a stronger
property: instead of having a line for each point of the set, one
can have, for each point of the set and for each angular aperture
at that point (no matter how narrow, no matter what its orientation),
continuum many such lines through that point.
My interest is along other lines. (Sorry!) Does there exist a set
that is categorically dense (i.e. has a first category complement,
relative to the unit square) having the property that for each of
its points there exists a line through that point .....? (I am not
particularly interested in whether Davies' stronger version has a
category analog.) Even better, does there exist a set having this
"linear accessibility" property whose complement (relative to the
unit square) is both measure zero and first category?
Of course, I suppose one could also ask if such a set exists which
has a sigma-porous complement, Hausdorff (or other) dimension less
than 2 complement, etc. But my primary interest is actually just in
the "complement is first category" version.
by
Juris Steprans
(Juris.Steprans@mathstat.yorku.ca)
Comment added June 30, 1997.
I was able to partially answer a Dave Renfro's
question in the Set Theoretic Analysis Page who
asked whether there is a category
version of a Nikodym set. I can show that there is a comeagre set X in the
plane such that for each x in X there is a closed unit line segment L
such that L\cap X = {x}. The original question was for L being a line
rather than line segment. (However, I think that Nikodym's original
construction gave a full measure set with the line segment property I
mentioned.) In any case, I suspect that the construction can be imporoved
to yield the full result so I will continue to think about it.
Juris Steprans
Department of Mathematics
York University
Toronto, Ontario
Canada M3J 1P3
(416) 736-5250 (ext. 33952)
Update added September 30, 1997
After writing the solution mentioned above, it was pointed out to me that the same
results (although with different arguments) have been obtained by
Bhagamil & Humke. A good reference that also points to other sources is
the paper by Paul Humke in J. London Math Soc. (2), 14 (1976), 245-248.
Juris Steprans
Copyright © 1997 by Topology Atlas. All rights reserved.