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Kakutani Partition

A Kakutani partition is an interval subdivision obtained by repeatedly splitting every longest current interval in a fixed ratio. Fix 0<alpha<1 and begin with the unit interval [0,1]. At each stage, replace every longest interval [u,v] by the two intervals [u,u+alpha(v-u)] and [u+alpha(v-u),v], leaving all shorter intervals unchanged. For alpha=1/2, stage n consists of 2^n equal intervals of length 2^(-n).

Kakutani's equidistribution theorem states that choosing one point in the interior of each interval gives finite point sets whose relative frequencies in subintervals approach the corresponding lengths (Kakutani 1976, Smilansky 2020).

Smilansky (2020) generalizes the largest-first construction and its equidistribution result to multiscale substitution tiling schemes, including right triangle refinements.

See also

Equidistributed Sequence, Interval, Kakutani's Equidistribution Theorem, Substitution Tiling, Uniform Distribution Theory

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References

Kakutani, S. "A Problem of Equidistribution on the Unit Interval [0,1]." In Measure Theory: Proceedings of the Conference Held at Oberwolfach, 15-21 June, 1975 (Ed. A. Bellow and D. Kölzow). Berlin, Germany: Springer-Verlag, pp. 369-375, 1976. https://doi.org/10.1007/BFb0081068.Smilansky, Y. "Uniform Distribution of Kakutani Partitions Generated by Substitution Schemes." Israel J. Math. 240, 667-710, 2020. https://doi.org/10.1007/s11856-020-2075-z.

Cite this as:

Weisstein, Eric W. "Kakutani Partition." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KakutaniPartition.html

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