Nikodym set explained
In mathematics, a Nikodym set is a subset of the unit square in
with complement of
Lebesgue measure zero (i.e. with an area of 1), such that, given any point in the set, there is a straight line that only intersects the set at that point.
[1] The existence of a Nikodym set was first proved by
Otto Nikodym in 1927. Subsequently, constructions were found of Nikodym sets having continuum many exceptional lines for each point, and
Kenneth Falconer found analogues in higher dimensions.
[2] Nikodym sets are closely related to Kakeya sets (also known as Besicovitch sets).
The existence of Nikodym sets is sometimes compared with the Banach–Tarski paradox. There is, however, an important difference between the two: the Banach–Tarski paradox relies on non-measurable sets.
Mathematicians have also researched Nikodym sets over finite fields (as opposed to
).
[3] Notes and References
- Book: Bogachev. Vladimir I.. Measure Theory. 2007. Springer Science & Business Media. 9783540345145. 67. en.
- Falconer. K. J.. Sets with Prescribed Projections and Nikodym Sets. Proceedings of the London Mathematical Society. s3-53. 1. 48–64. 10.1112/plms/s3-53.1.48. 1986.
- Book: Graham. Ronald L.. Ronald Graham. Steve Butler (mathematician). Nešetřil. Jaroslav. Butler. Steve. The Mathematics of Paul Erdős I. 2013. Springer Science & Business Media. 9781461472582. 496. en.
