Pollington 1979 density sequence n k xi

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theorem: Pollington's 1979 theorem that every sequence of positive numbers with consecutive ratios at least a fixed alpha above 1 admits, for each s_0 below 1, a positive beta and a set of multipliers of Hausdorff dimension at least s_0 whose fractional parts along the sequence all lie in [beta, 1 - beta]; with its corollary that the exceptional set has dimension 1, a solution of Erdős's lacunary density question independent of de Mathan's.

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A. D. Pollington, On the density of sequence $\{n_k\xi\}$, Illinois J. Math. 23 (1979), no. 4, 511--515; DOI 10.1215/ijm/1256047933. Received 14 February 1979. The site's key Po79b.

The copy read for this card is the journal's scan of the five printed pages (pp. 511--515; PDF p. $n$ is printed p. $510+n$) with an OCR text layer that garbles the formulas, so every statement below was read on the rendered page images. Provenance: retrieved from the journal's open back file on Project Euclid, https://projecteuclid.org/journalArticle/Download?urlid=10.1215%2Fijm%2F1256047933 (HTTP 200, application/pdf, one request, after the DOI resolved to the article's landing page); 293,566 bytes. The foot of printed p. 511 carries the notice "© 1979 by the Board of Trustees of the University of Illinois"; the journal's article page on Project Euclid (reached from https://doi.org/10.1215/ijm/1256047933, read 2026-10-02) shows an "Open Access" icon and no copyright line, Creative Commons license or rights statement, so no open license is named and the printed notice gives the term reserved.

Read status: claims checked for the Theorem, the Corollary and the introduction's statement of Erdős's question (read clause by clause on the page image of printed p. 511), and for the uncountability sentence on printed p. 514; the proof (pp. 511--515) was read for structure and not checked.

Contents

• Introduction (p. 511): the paper opens by attributing the question to Erdős's Problems and results in Diophantine approximations II in [2], Répartition modulo 1, Lecture Notes in Mathematics 475, Springer, 1975, and poses it as follows (p. 511): "Given a sequence of integers $n_1<n_2<n_2\cdots$ [sic] satisfying $n_{k+1}/n_k\ge\alpha>1$, $k=1,2,\ldots$, is it true that there always exists an irrational $\xi$ for which the sequence $\{n_k\xi\}$ is not everywhere dense?", where $\{x\}$ is the fractional part. Strzelecki [5] had shown the conclusion, with $\{t_k\xi\}\in[\beta,1-\beta]$ for some $\beta>0$, for real sequences with ratio $\alpha\ge5^{1/3}$. The paper presents its Theorem as "a complete answer to the question of Erdös" (p. 511).
• The Theorem (p. 511): for a sequence $(t_n)$ of positive numbers with $q_n=t_{n+1}/t_n\ge\alpha>1$ for all $n$ (1), and any real $0<s_0<1$, there are $\beta=\beta(\alpha,s_0)>0$ and a set $T$ whose Hausdorff dimension is at least $s_0$ on which $\{t_k\xi\}\in[\beta,1-\beta]$ for every $k$ (2). Corollary (p. 511): the exceptional set, the $\xi$ whose sequence $\{t_k\xi\}$ fails to be dense in $[0,1]$, has Hausdorff dimension 1. The paper notes (p. 511) that de Mathan [3], [4] had recently and independently obtained a similar result.
• Proof (pp. 511--515): a reduction to $q_n\le\alpha^2$ by inserting terms; a lemma constructing closed intervals $[a_n,b_n]$ with no integer interior points and the conditions (A)--(D), so that the intervals $[a_n/t_n,b_n/t_n]$ are nested and $\xi$ in their intersection satisfies (2) with $\beta=\frac12N^{-r}\varepsilon$ in the paper's notation (4a); the construction offers two disjoint intervals to choose from at each stage, which gives uncountably many such $\xi$ (p. 514); Eggleston's theorem [1] on sets defined by nested families of intervals gives the Hausdorff dimension (pp. 514--515).
• References (p. 515): Eggleston 1951--52; Erdős, Répartition modulo 1, LNM 475 (1975); de Mathan, C. R. Acad. Sci. Paris Sér. A 287 (1978), 277--279, and "numbers contravening a condition in density modulo 1, to appear"; Strzelecki, Canad. Math. Bull. 18 (1975), 727--738.

Compiled scope

All five pages were rendered and read; the Theorem, Corollary and the question are compiled as statements with a proof pointer. No step of the proof was checked and nothing here is independently reviewed. The Theorem does not itself mention irrationality; the paper's uncountability remark supplies it (an uncountable set of reals contains irrationals), which the consuming pages state as an authored line.

Bears on. #464, as one of the two independent solutions the site names (p. 511 credits de Mathan, whose 1978 Comptes Rendus note is its [3], with a similar result): the Theorem with $t_k=n_k$ and $\alpha=1+\epsilon$ gives irrational $\theta$ with $\{\theta n_k\}\in[\beta,1-\beta]$ for all $k$, hence not dense modulo $1$ and $\inf_k\|\theta n_k\|\ge\beta>0$, the corrected Statement of that page (the site's wording is trivially true); the introduction quotes Erdős's 1975 question with its irrational clause, and the reference list places that question in the LNM 475 volume the site cites as Er75i. #894, where the same separation gives, through Katznelson's reduction, a proper coloring of the lacunary difference graph with $\lceil\beta^{-1}\rceil$ colors, a second first-hand proof of finiteness without an explicit dependence on $\epsilon$. Katznelson's paper is filed as katznelson_2001_chromatic_numbers_cayley_graphs_z_recurrence; the reduction is his proof of Theorem 1.1 on printed p. 212 (PDF p. 2), which divides the circle into $M$ equal arcs with $M\varepsilon>1$ and colors $n$ by the arc containing $n\alpha$, read there in the text layer and paged on theorem_1_1.

Results.

• Theorem (p. 511): $\{t_k\xi\}\in[\beta,1-\beta]$ for all $k$ and all $\xi$ in a set of Hausdorff dimension at least $s_0$, for every lacunary sequence of positive numbers with ratio at least $\alpha>1$; with the Corollary that the exceptional set has Hausdorff dimension 1.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.