Problem 464

Let $A=\{n_1<n_2<\cdots\}\subset \mathbb{N}$ be a lacunary sequence (so there exists some $\epsilon>0$ with $n_{k+1}\geq (1+\epsilon)n_k$ for all $k$ ). Must there exist an irrational $\theta$ such that

\{ \|\theta n_k\| : k\geq 1\}

is not dense in $[0,1]$ (where $\| x\|$ is the distance to the nearest integer)?

Second statement

\{ \{\theta n_k\} : k\geq 1\}

is not dense in $[0,1]$ (where $\{x\}$ is the fractional part of $x$ )?

Source: erdosproblems.com/464

Number theory

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. The site's label PROVED (LEAN) describes the site's wording, which is trivially true (Notes), and is right for the corrected Statement as well; the suffix is a catalog label explained under Formalization below. Pollington's Theorem [Po79b] (Illinois J. Math. 23 (1979), refereed) gives, for every sequence of positive numbers with consecutive ratios at least $\alpha>1$ , a $\beta>0$ and a set of $\xi$ of positive Hausdorff dimension, uncountable by the paper's own remark, with $\{t_k\xi\}\in[\beta,1-\beta]$ for all $k$ ; an uncountable set of reals contains irrational numbers (the rationals are countable), so an irrational $\theta$ with $\inf_k\|\theta n_k\|\ge\beta>0$ exists (one authored line). De Mathan's independent solution [dM80] (Acta Math. Acad. Sci. Hungar. 36 (1980), refereed) is its Corollary 1 (p. 237): for every sequence of positive reals with consecutive ratios at least $\lambda>1$ and every interval $[a,b]$ , the $x\in[a,b]$ with $(q_nx)$ not everywhere dense mod 1 form a set of Hausdorff dimension 1, and the proof of its Theorem 1 concludes (p. 241) that the $x$ for which $(q_nx)$ does not have $0$ as a point of accumulation mod 1, so that $\|q_nx\|$ stays above some $\varepsilon>0$ for all but finitely many $n$ , also form a set of dimension 1; the same uncountability line supplies an irrational $\theta$ in that set, and for it the finitely many excepted terms have $\|\theta n_k\|>0$ , so $\inf_k\|\theta n_k\|>0$ (two authored lines). Katznelson's Theorem 1.2 [Ka01] (Combinatorica 21 (2001), refereed) gives the separation again, and his Claim 2 gives the set of multipliers with a positive separation Hausdorff dimension 1, so the same uncountability line supplies an irrational $\theta$ . Pollington (p. 511), Katznelson (p. 212), Erdős's 1982 restatement (p. 63) and the site attest the independence of the two original solutions; Peres and Schlag (p. 2) credit both without saying so. The three results are the accepted claim pages Pollington 1979, de Mathan 1980 and Katznelson 2001. The quantitative record: de Mathan and Pollington give $\inf_k\|\theta n_k\|\gg\epsilon^4/\log(1/\epsilon)$ (as Peres and Schlag report it; de Mathan's paper prints no bound in terms of $\lambda-1$ , and its displayed choices give a separation of order $\epsilon^4/\log^4(1/\epsilon)$ , a filing observation recorded on its card), Katznelson's footnote 2 prints $\varepsilon(\rho)> (\rho-1)^2\log^{-2}(\rho-1)$ for ratio $\rho$ close to $1$ , of order $\epsilon^2/\log^2(1/\epsilon)$ , where Peres and Schlag report $\gg\epsilon^2/\log(1/\epsilon)$ for it, Akhunzhanov and Moshchevitin remove the logarithm from the quoted form (as Peres and Schlag and Dubickas report it), Dubickas's Theorem 1 gives $\|\xi t_n\|\ge1/(9(r+2)^2)$ for ratio at least $1+r^{-1}$ , of order $\epsilon^2$ , and Peres and Schlag's Theorem 1.1 gives $\gg\epsilon/\log(1/\epsilon)$ for $0<\epsilon<1/4$ , sharp up to the logarithm. The Dubickas and Peres--Schlag theorems produce a positive real $\xi$ or a $\theta\in(0,1)$ and do not assert irrationality, and Akhunzhanov and Moshchevitin's bound is known only as those two papers report it, with no irrational multiplier; these three results settle no instance of the question and have no claim pages. The irrationality clause rests on Pollington's, de Mathan's and Katznelson's papers, each through the uncountability of its dimension-one set.

Notes

The site's wording is true at every instance for a trivial reason: the distance $\|x\|$ to the nearest integer never exceeds $1/2$ , so the displayed set lies in $[0,1/2]$ for every real $\theta$ and is never dense in $[0,1]$ ; any irrational $\theta$ , such as $\sqrt2$ for $A=\{2^k\}$ , answers it yes, for every sequence $A$ , without any use of lacunarity. The change replaces the distances $\|\theta n_k\|$ in the display by the fractional parts $\{\theta n_k\}$ , and the closing "(where $\| x\|$ is the distance to the nearest integer)" by "(where $\{x\}$ is the fractional part of $x$ )"; nothing else changes. The evidence is Erdős's own statements of the question. The 1975 chapter [Er75i], printed p. 96, asks whether "there always is an irrational $\alpha$ for which the sequence $(n_k\alpha)$ is not everywhere dense", where $(x)$ is the chapter's notation for the fractional part (its § 3, p. 91), and the 1982 survey [Er82e], printed p. 63, restates it as "there is always an irrational $\alpha$ for which the fractional part of $n_k\alpha$ is not everywhere dense". Pollington's introduction (p. 511) quotes the question in the same form, with $\{x\}$ the fractional part, and de Mathan (p. 237) states it as density modulo $1$ . The defect is the site's: no statement of Erdős uses the distance to the nearest integer. The trivial truth of the site's wording was pointed out by the AI system Aristotle while formalizing de Mathan's argument, as the forum account JoshuaB reported in the site's thread on 21 June 2026 (the comment), proposing a modulo-one wording; the formal-conjectures file at the commit linked under Formalization says in its formalization notes that the printed wording "would be vacuously true". These observations are credited here and settle nothing about the corrected Statement.

Er75i: Erdős, P., Problems and results on diophantine approximations (II). In Répartition modulo 1 (Actes du Colloque de Marseille-Luminy 1974), Lecture Notes in Mathematics 475, Springer, 1975, pp. 89--99. The site's reference text (its reference service) names the volume alone ("Répartition modulo 1. (1975), iv+258"); the other sources identify Erdős's chapter in it ([Du06], reference [14], with pp. 89--99; [Er82e], reference list on p. 63, with pp. 89--97, two pages short of the chapter's printed pp. 89--99; Pollington's introduction names the same paper). The chapter, printed pp. 89--99, prints neither the volume's title nor its year. Printed p. 96: the question quoted below, with the irrational clause and the fractional-part notation. Library home: Erdős 1975.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980). Printed p. 18: the passage below, crediting Pollington. Library home: Erdős and Graham 1980.
Er82e: Erdős, P., Some of my favourite problems which recently have been solved. (1982), 59--79 (the site's text). Printed p. 63: Erdős's restatement and announcement. Library home: Erdős 1982.
Po79b: Pollington, A. D., 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 Theorem and Corollary (p. 511); the uncountability remark (p. 514). Open access in the journal's back file. Library home: Pollington 1979.
dM80: de Mathan, B., Numbers contravening a condition in density modulo 1. Acta Math. Acad. Sci. Hungar. 36 (1980), no. 3--4, 237--241, doi:10.1007/BF01898138 (received 28 November 1978). Theorem 1 and Corollaries 1--2 (p. 237), the separation in the proof (p. 238), the conclusion and the added-in-proof note crediting Pollington (p. 241). Cited as "to appear" by Pollington ([4], with the 1978 Comptes Rendus note [3], not read) and by Peres and Schlag as the other original solution. Library home: de Mathan 1980.
Ka01: Katznelson, Y., Chromatic numbers of Cayley graphs on $\mathbb Z$ and recurrence. Combinatorica 21 (2001), no. 2, 211--219, doi:10.1007/s004930100019 (received 7 February 2000). Theorem 1.2, Claims 1--2 and footnote 2 (p. 212), and the attribution of the question to [Er75i] and of its answers to de Mathan and Pollington (p. 212). Library home: Katznelson 2001.
AkMo04: Akhunzhanov, R. K. and Moshchevitin, N. G., On the chromatic number of a distance graph associated with a lacunary sequence. Dokl. Akad. Nauk 397 (2004), 295--296. Not read; quoted from [PeSc10] p. 2 and [Du06] p. 137 (where the bound $\chi\le2^7r^2$ for $r\ge3$ is attributed to it).
Du06: Dubickas, A., On the fractional parts of lacunary sequences. Math. Scand. 99 (2006), no. 1, 136--146, doi:10.7146/math.scand.a-15004 (received 20 July 2005). Theorem 1 (p. 136), the chromatic bound (p. 137), Corollaries 2--3 (p. 137). Library home: Dubickas 2006.
PeSc10: Peres, Y. and Schlag, W., Two Erdős problems on lacunary sequences: chromatic number and Diophantine approximation. Bull. Lond. Math. Soc. 42 (2010), no. 2, 295--300, doi:10.1112/blms/bdp126; arXiv:0706.0223v1 (1 June 2007). Problem B (p. 1), Theorem 1.1 and the history (p. 2). Library home: Peres and Schlag 2010.
St25: Stefanescu, R., The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation. Adv. Math. 461 (2025), 110062, doi:10.1016/j.aim.2024.110062 (Crossref record accessed; the paper not read). A lead on the quantitative question, named with its identifier.

Problem 464

Second statement

Status

Notes

Proof claims

Discussion