Proved. Theorem 1.1 of Peres and Schlag [PeSc10], cited from the arXiv version and published in the Bulletin of the London Mathematical Society in 2010 (refereed; page references follow the preprint), gives, for every lacunary $S$ with ratio at least $1+\epsilon$, $0<\epsilon<1/4$, a $\theta\in(0,1)$ with $\inf_j\|\theta n_j\|>c\epsilon |\log\epsilon|^{-1}$ and hence, by Katznelson's reduction (the proof of his Theorem 1.1, [Ka01] p. 212; restated in the same paper), a proper coloring of the graph with at most $1+c^{-1}\epsilon^{-1}|\log\epsilon|$ colors; the site's $\ll\epsilon^{-1} \log(1/\epsilon)$ bound is this. The same paper gives a self-contained elementary coloring with $4^K$ colors, $K=\lceil2\epsilon^{-1}\rceil$, on its p. 3, and shows that the power of $\epsilon$ in the bound cannot be improved. The earlier solutions of the Diophantine question behind the reduction (de Mathan and Pollington, 1979--80) are the site's account of Problem 464; Pollington's theorem gives a second route to finiteness through the reduction, without an explicit dependence on $\epsilon$. Katznelson's own Theorem 1.1 (p. 211), "If $\Lambda$ is lacunary then $\chi(\Lambda)<\infty$", is the original answer, proved from his Theorem 1.2 by the reduction and again, elementarily, with $5^d$ colors where $\rho^d\ge5$ (p. 212). The frontmatter standing is derived from the claim pages: Peres and Schlag's theorem (claim page (Peres and Schlag, 2007)) and Katznelson's (claim page (Katznelson, 2001)) are accepted full claims on their refereed publications and on the site's commentary, which credits both papers.