  Let $\Sigma$ be a countable alphabet. For $r\geq 1$, an infinite sequence $s$
with characters from $\Sigma$ is called $r$-quasi-regular, if for each
$\sigma\in\Sigma$ the ratio of the longest to shortest interval between
consecutive occurrences of $\sigma$ in $s$ is bounded by $r$. In this paper, we
answer a question asked by Kempe, Schulman, and Tamuz, and prove that for any
probability distribution $\mathbf{p}$ on a finite alphabet $\Sigma$, there
exists a $2$-quasi-regular infinite sequence with characters from $\Sigma$ and
density of characters equal to $\mathbf{p}$. We also prove that as
$\left\lVert\mathbf{p}\right\rVert_\infty$ tends to zero, the infimum of $r$
for which $r$-quasi-regular sequences with density $\mathbf{p}$ exist, tends to
one. This result has a corollary in the Pinwheel Problem: as the smallest
integer in the vector tends to infinity, the density threshold for Pinwheel
schedulability tends to one.
