Fraenkel’s partition theorem

Fraenkel’s partitionMathworldPlanetmathPlanetmath theorem is a generalizationPlanetmathPlanetmath of Beatty’s Theorem. Set


We say that two sequencesMathworldPlanetmath partition ={1,2,3,} if the sequences are disjoint and their union is .

Fraenkel’s Partition Theorem: The sequences B(α,α) and B(β,β) partition N if and only if the following five conditions are satisfied.

  1. 1.


  2. 2.


  3. 3.


  4. 4.

    If α is irrational, then α+β=0 and kα+α for 2k.

  5. 5.

    If α is rational (say q is minimalPlanetmathPlanetmath with qα), then 1qα+α and qα+qβ=1.



] Aviezri S. Fraenkel, The bracket function and complementary sets of integers, Canad. J. Math. 21 (1969), 6–27. 38:3214


] Kevin O’Bryant, Fraenkel’s partition and Brown’s decomposition,

Title Fraenkel’s partition theorem
Canonical name FraenkelsPartitionTheorem
Date of creation 2013-03-22 13:40:09
Last modified on 2013-03-22 13:40:09
Owner Kevin OBryant (1315)
Last modified by Kevin OBryant (1315)
Numerical id 6
Author Kevin OBryant (1315)
Entry type Theorem
Classification msc 11B83
Synonym Fraenkel’s theorem
Related topic BeattySequence
Related topic BeattysTheorem
Related topic DataStream
Related topic WideraInterlaceAndDeinterlace