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arXiv · 1909.13320

Quasi-Regular Sequences

Joshua Frisch·Wade Hann-Caruthers·Pooya Vahidi Ferdowsi
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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.

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Joshua Frisch, Wade Hann-Caruthers, Pooya Vahidi Ferdowsi. 2019-09-29. Quasi-Regular Sequences. https://arxiv.org/abs/1909.13320

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