Quasi-Regular Sequences

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.