  Given a real number $\alpha \in (0,1)$, we define the Webster sequence of
density $\alpha$ to be $W_\alpha = (\lceil(n-1/2) /
\alpha\rceil)_{n\in\mathbb{N}}$, where $\lceil x \rceil$ is the ceiling
function. It is known that if $\alpha$ and $\beta$ are irrational with $\alpha
+ \beta = 1$, then $W_\alpha$ and $W_\beta$ partition $\mathbb{N}$. On the
other hand, an analogous result for three-part partitions does not hold: There
does not exist a partition of $\mathbb{N}$ into sequences $W_\alpha, W_\beta,
W_\gamma$ with $\alpha, \beta, \gamma $ irrational. In this paper, we consider
the question of how close one can come to a three-part partition of
$\mathbb{N}$ into Webster sequences with prescribed densities $\alpha, \beta,
\gamma $. We show that if $\alpha, \beta, \gamma $ are irrational with $\alpha
+ \beta +\gamma = 1$, there exists a partition of $\mathbb{N}$ into sequences
of densities $\alpha, \beta, \gamma$, in which one of the sequences is a
Webster sequence and the other two are "almost" Webster sequences that are
obtained from Webster sequences by perturbing some elements by at most 1. We
also provide two efficient algorithms to construct such partitions. The first
algorithm outputs the $n$th element of each sequence in $O(1)$ time and the
second algorithm gives the assignment of the $m$th positive integer to a
sequence in $O(1)$ time. We show that the results are best-possible in several
respects. Moreover, we describe applications of these results to apportionment
and optimal scheduling problems.
