Quasi-Regular Sequences

TL;DR

This paper proves the existence of 2-quasi-regular sequences with any given character distribution and explores how the regularity parameter approaches one as character densities decrease, impacting scheduling theory.

Contribution

It establishes the existence of 2-quasi-regular sequences matching any finite alphabet distribution and links the regularity parameter to character densities, answering an open question.

Findings

Existence of 2-quasi-regular sequences for any distribution on finite alphabet.

As maximum character probability decreases, the regularity parameter approaches one.

Implication for the Pinwheel Problem and schedulability thresholds.

Abstract

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.