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

Department of Mathematics

University of Illinois

1409 W. Green St.

Urbana, IL 61801

USA

Junxian Li

Mathematisches Institut

Bunsenstraße 3–5

D-37073 Göttingen

Germany

Yun Xie

Department of Statistics

University of Washington

Box 354322

Seattle, WA 98195

USA

**Abstract:**

Given 0 < α < 1, the Beatty sequence of density α
is the sequence *B*_{α} = (⌊ n/α⌋)
_{n ∈ N}.
Beatty's theorem states that if α, β are irrational
numbers with α+β = 1, then the Beatty sequences
*B*_{α} and *B*_{β} 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
*B*_{α},
*B*_{β},
*B*_{γ}
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.

(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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