Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.6

Almost Beatty Partitions

A. J. Hildebrand and Xiaomin Li
Department of Mathematics
University of Illinois
1409 W. Green St.
Urbana, IL 61801

Junxian Li
Mathematisches Institut
Bunsenstraße 3–5
D-37073 Göttingen

Yun Xie
Department of Statistics
University of Washington
Box 354322
Seattle, WA 98195


Given 0 < α < 1, the Beatty sequence of density α is the sequence Bα = (⌊ n/α⌋) n ∈ N. Beatty's theorem states that if α, β are irrational numbers with α+β = 1, then the Beatty sequences Bα and Bβ partition the positive integers; that is, each positive integer belongs to exactly one of these two sequences. On the other hand, Uspensky showed that this result breaks down completely for partitions into three (or more) sequences: there does not exist a single triple (α, β, γ) such that the Beatty sequences Bα, Bβ, Bγ partition the positive integers.

In this paper we consider the question of how close we can come to a three-part Beatty partition by considering "almost" Beatty sequences, that is, sequences that represent small perturbations of an "exact" Beatty sequence. We first characterize all cases in which there exists a partition into two exact Beatty sequences and one almost Beatty sequence with given densities, and we determine the approximation error involved. We then give two general constructions that yield partitions into one exact Beatty sequence and two almost Beatty sequences with prescribed densities, and we determine the approximation error in these constructions. Finally, we show that in many situations these constructions are best-possible in the sense that they yield the closest approximation to a three-part Beatty partition.

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(Concerned with sequences A000201 A003144 A003145 A003146 A003623 A004919 A004976 A158919 A277722 A277723 A277728.)

Received October 5 2018; revised version received July 3 2019. Published in Journal of Integer Sequences, July 7 2019.

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