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Katznelson proves, as Theorem 1.2 of his 2001 paper (p. 212), that for every $\rho>1$ there is $\varepsilon=\varepsilon(\rho)>0$ such that every lacunary $\Lambda=\{\lambda_j\}\subset\mathbb N$ with parameter $\rho$, that is with $\lambda_{j+1}/\lambda_j\ge\rho$, has some $\alpha\in\mathbb T$ with $\|\lambda\alpha\|>\varepsilon$ for all $\lambda\in\Lambda$; his footnote 2 gives $\varepsilon(\rho)>(\rho-1)^2\log^{-2}(\rho-1)$ for $\rho$ close to $1$. The proof establishes two claims, and Claim 2 (p. 212) states that for a finite union $\Lambda$ of lacunary sequences the set $A(\Lambda)={\alpha:\text{some }\varepsilon>0\text{ has }|\lambda\alpha|> \varepsilon\text{ for all }\lambda\in\Lambda}$ has Hausdorff dimension $1$.

For Problem 464 take $\Lambda=A$ and $\rho=1+\epsilon$. The set $A(\Lambda)$ has Hausdorff dimension $1$, so it is uncountable and contains an irrational $\theta$ (the rationals are countable); such a $\theta$ has $\inf_k\|\theta n_k\|>0$, so the fractional parts $\{\theta n_k\}$ avoid a neighborhood of $0$ modulo $1$ and $(\theta n_k)$ is not dense modulo $1$, which is the problem page's corrected Statement. Theorem 1.2 alone produces $\alpha\in\mathbb T$ without an irrationality clause; the irrational multiplier comes from Claim 2 by this authored line. The paper records on p. 212 that the question was raised by Erdős in his 1975 chapter and answered independently by de Mathan and Pollington, whose solutions have their own pages, de Mathan 1980 and Pollington 1979.

The paper's library home is Katznelson 2001, with a compiled page for Theorem 1.2; the statements are taken first-hand from the paper, the proof of the claims for $\rho$ close to $1$ (p. 213) is followed for structure only, and nothing here is independently reviewed.

Acceptance. The paper is refereed: Y. Katznelson, Chromatic numbers of Cayley graphs on $\mathbb Z$ and recurrence, Combinatorica 21, no. 2 (2001), 211--219, received 7 February 2000. The site's curator credits the paper only with an improved separation bound, not with the solution, so no review by the site is listed.

Depends on. No page of this wiki. The proof is self-contained in the paper.

Date. The page is dated by the first day of the issue month, April 2001, since the paper's first posting carries no finer date.