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Statement

Quoted from p. 511 (the paper's own theorem, unnumbered; p. 514 also states Eggleston's theorem):

Theorem. "Let $(t_n)$ be a sequence of positive numbers such that $q_n=t_{n+1}/t_n\ge\alpha>1$ for $n=1,2,\ldots$ (1) and let $s_0$ be a real number $0<s_0<1$ then there exists a real number $\beta=\beta(\alpha,s_0)>0$ and a set $T$ of Hausdorff dimension at least $s_0$ such that if $\xi\in T$ then $\{t_k\xi\}\in[\beta,1-\beta]$ for $k=1,2,\ldots$ (2)."

Corollary. "The set of numbers $\xi$ such that $\{t_k\xi\}$ is not dense in the unit interval has Hausdorff dimension 1."

Here $\{x\}$ is the fractional part of $x$. The paper adds on the same page: "A similar result has recently been obtained independently by B. de Mathan [3], [4]." On p. 514 the proof notes that "there are uncountably many such $\xi$", since each stage of the construction offers two disjoint choices of interval. The Theorem does not mention irrationality; that an irrational $\xi$ exists follows from uncountability, since the rationals are countable (an elementary line the consuming pages state as authored). For a lacunary sequence of positive integers $n_k$ with $n_{k+1}\ge(1+\epsilon)n_k$ the Theorem applies with $t_k=n_k$ and $\alpha=1+\epsilon$, and (2) gives $\|\xi n_k\|\ge\beta$ for all $k$.

Source. A. D. Pollington, On the density of sequence $\{n_k\xi\}$, Illinois J. Math. 23 (1979), no. 4, 511--515; the Theorem and Corollary on printed p. 511 (PDF p. 1 of the journal scan), the uncountability sentence on printed p. 514 (PDF p. 4), read on the rendered page images (the OCR text layer garbles the formulas). The edition read is identified in the source digest.

Read depth. Claims checked: the Theorem, the Corollary, the sentence on de Mathan and the uncountability sentence were read clause by clause on the page images. The proof was read for structure and not checked.

Proof pointer

Pp. 511--515. The paper first reduces to $q_n\le\alpha^2$ by inserting terms between $t_k$ and $t_{k+1}$ where the ratio is large (p. 511). Choosing $r$ with $\alpha^r-(r+2)>\alpha^{rs_0}$ (3), $N=\alpha^2$ and $\varepsilon=N^{-r}(r+1)^{-1}$ (4), it takes $\beta=\frac12N^{-r}\varepsilon$ (4a) and constructs, by a lemma proved in blocks of $r$ indices (pp. 512--514), closed intervals $[a_n,b_n]$ with no integer interior points and $q_na_n\le a_{n+1}<b_{n+1}\le q_nb_n$, so that by (1) the intervals $[a_n/t_n,b_n/t_n]$ are nested; any $\xi$ in their intersection satisfies $a_m+\beta\le t_m\xi\le b_m-\beta$ for all $m$, hence (2). At each stage there are at least two disjoint admissible intervals, so the set of such $\xi$ is uncountable, and Eggleston's theorem on sets defined by nested families of intervals (stated on p. 514) gives Hausdorff dimension at least $s_0$ (pp. 514--515). Not reconstructed here.

Dependencies

Eggleston's theorem (H. G. Eggleston, Sets of fractional dimension which occur in some problems of number theory, Proc. London Math. Soc. 54 (1951--52), 42--93; the paper's [1]), cited and stated, not proved, in the paper; otherwise self-contained.

Bears on

• Problem 464: a solution of the corrected Statement (fractional parts not dense modulo $1$, with an irrational multiplier), independent of de Mathan's, whose 1978 Comptes Rendus note p. 511 cites as [3]; the separation $\|\xi n_k\|\ge\beta$ is the qualitative form of the bounds Katznelson, Dubickas and Peres and Schlag later quantified.
• Problem 894: through Katznelson's reduction, the separation yields a proper coloring of the lacunary difference graph with $\lceil\beta^{-1}\rceil$ colors; the paper does not make $\beta$'s dependence on $\alpha$ explicit.