Quasi-Regular Sequences

Abstract

Let $Σ$ be a countable alphabet. For $r\geq 1$, an infinite sequence $s$ with characters from $Σ$ is called $r$-quasi-regular, if for each $σ\inΣ$ the ratio of the longest to shortest interval between consecutive occurrences of $σ$ 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 $Σ$, there exists a $2$-quasi-regular infinite sequence with characters from $Σ$ 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.