Nikodym set

In mathematics, the Nikodym set is a subset of the unit square in $\mathbb {R} ^{2}$ such that, given any point in the set, there is a straight line that only intersects the set at that point. The existence of the Nikodym set was first proved by Otto Nikodym in 1927. Subsequently, constructions were found for the Nikodym set and Kenneth Falconer found analogues in higher dimensions.^[1]

The Nikodym set is a Borel set $N\subseteq [0,1]\times [0,1]$ with area (more precisely, Lebesgue measure) equal to 1 such that for all $n\in N$ , there exists a straight line $l_{n}$ in $\mathbb {R} ^{2}$ such that $l_{n}\cap N=\{n\}$ .^[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 $\mathbb {R}$ ).^[3]

References

↑ Falconer, K. J. "Sets with Prescribed Projections and Nikodym Sets". Proceedings of the London Mathematical Society. s3-53 (1): 48–64. doi:10.1112/plms/s3-53.1.48.
↑ Bogachev, Vladimir I. (2007). Measure Theory. Springer Science & Business Media. p. 67. ISBN 9783540345145.
↑ Graham, Ronald L.; Nešetřil, Jaroslav; Butler, Steve (2013). The Mathematics of Paul Erdős I. Springer Science & Business Media. p. 496. ISBN 9781461472582.

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[1] Falconer, K. J. "Sets with Prescribed Projections and Nikodym Sets". Proceedings of the London Mathematical Society. s3-53 (1): 48–64. doi:10.1112/plms/s3-53.1.48.

[2] Bogachev, Vladimir I. (2007). Measure Theory. Springer Science & Business Media. p. 67. ISBN 9783540345145.

[3] Graham, Ronald L.; Nešetřil, Jaroslav; Butler, Steve (2013). The Mathematics of Paul Erdős I. Springer Science & Business Media. p. 496. ISBN 9781461472582.