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Pollington proves, as the Theorem of his 1979 paper, that if $(t_n)$ is a sequence of positive numbers with $t_{n+1}/t_n\ge\alpha>1$ for all $n$ and $0<s_0<1$, then there are $\beta=\beta(\alpha,s_0)>0$ and a set $T$ of Hausdorff dimension at least $s_0$ such that every $\xi\in T$ has $\{t_k\xi\}\in[\beta,1-\beta]$ for all $k$; the Corollary states that the $\xi$ for which $\{t_k\xi\}$ is not dense in the unit interval form a set of dimension $1$, and the proof notes that the nested-interval construction offers two disjoint choices at each stage, so that there are uncountably many such $\xi$.

For Problem 464 take $t_k=n_k$ and $\alpha=1+\epsilon$. Every $\theta\in T$ has $\|\theta n_k\|\ge\beta$ for all $k$, so the sequence $(\theta n_k)$ is not dense modulo $1$, which is the problem page's corrected Statement, the question in the form Erdős posed it in 1975 and restated in 1982; the site's wording, that the set of distances $\|\theta n_k\|$ is not dense in $[0,1]$, holds for every $\theta$ already because $\|x\|\le1/2$ (the problem page's Notes). The question asks for an irrational $\theta$: $T$ is uncountable and the rationals are countable, so an irrational $\theta\in T$ exists (the problem page's authored line). Pollington's introduction poses the question with the irrational clause and calls the Theorem a complete answer to it. De Mathan's independent solution has its own page; each paper credits the other.

The paper's library home is Pollington 1979, with a compiled page for the Theorem; the statements are taken first-hand from the paper, the proof, nested intervals and Eggleston's theorem for the dimension, is followed for structure only, and nothing here is independently reviewed.

Acceptance. The paper is refereed: A. D. Pollington, On the density of sequence $\{n_k\xi\}$, Illinois J. Math. 23, no. 4 (December 1979), 511--515, received 14 February 1979. Erdős announced in 1982 that de Mathan and Pollington had settled the problem independently, and Katznelson records the same; Peres and Schlag credit both papers. The site's curator, Thomas F. Bloom, marks Problem 464 proved and credits this paper, with de Mathan's, with the solution. The page is dated by the first day of the issue month, since the paper's first posting carries no finer date.