Almost Beatty Partitions

Abstract

Given $0<\alpha<1$, the Beatty sequence of density $\alpha$ is the sequence $B_{\alpha}=(\lfloor n/\alpha\rfloor)_{n\in\mathbb{N}}$. Beatty's theorem states that if $\alpha,\beta$ are irrational numbers with $\alpha+\beta=1$, then the Beatty sequences $B_{\alpha}$ and $B_{\beta}$ 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 $(\alpha,\beta,\gamma)$ such that the Beatty sequences $B_\alpha,B_\beta,B_\gamma$ 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.