Does there exist a Baire category version of a Nikodym set?

Dave Renfro

Recall that a Nikodym set (in the unit square of R2) 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.

A partial solution to the problem

Juris Steprans
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

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