De mathan 1980 numbers contravening condition density modulo 1

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corollary_1: De Mathan's Corollary 1 that for every sequence of positive reals with consecutive ratios at least a fixed lambda above 1 and every interval, the set of x in the interval for which (q_n x) is not everywhere dense mod 1 has Hausdorff dimension 1; the paper's answer to Erdős's question and the statement Problem 464 consumes.

corollary_2: De Mathan's Corollary 2 that for every real v and every interval [a, b], the set of x in [a, b] for which the sequence (v x^n) is not everywhere dense mod 1 has Hausdorff dimension 1.

theorem_1: De Mathan's Theorem 1 that a sequence of monotonic differentiable functions on an interval whose consecutive derivative ratios lie between lambda and mu, 1 < lambda <= mu, has a point x whose values are not everywhere dense mod 1, and, under a Lipschitz condition on the logarithms of the derivatives, that the set of such x has Hausdorff dimension 1.

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B. de Mathan, Numbers contravening a condition in density modulo 1, Acta Mathematica Academiae Scientiarum Hungaricae 36 (1980), no. 3--4, 237--241, DOI 10.1007/BF01898138 (the DOI is the publisher's, not printed on the pages); received 28 November 1978, with an "Added in proof" dated 8 April 1980 (p. 241); the author at the U.E.R. de Mathématiques et d'Informatique of the Université de Bordeaux I (p. 241). Cited as [dM80] on the problem page. Its two references (p. 241) are Erdős and Taylor, On the set of points of convergence of a lacunary trigonometric series, and the equidistribution properties of related sequences, Proc. London Math. Soc. (3) 7 (1957), 598--615, and Thomas, Dimension de Hausdorff, Bull. Soc. Math. France 37 (1974), 161--167 (Journées Arithmétiques, Grenoble, 1973). The edition read for this card is the publisher's version of record; the paper's earlier announcement, the 1978 Comptes Rendus note that Pollington's reference list names, was not compared.

The copy read for this card is the publisher's scan of the printed article: 5 pages, printed pp. 237--241 = PDF pp. 1--5 (printed p. $n$ is PDF p. $n-236$), a 2005 scan (the scan's metadata names a TIFF source and a June 2005 creation date) with an OCR text layer that locates passages and garbles the formulas (the subscripts, the inequality signs, the Greek letters and the interval notation), so every statement below was read on the rendered page images. Provenance: the copy was obtained from the publisher on 2026-09-22 as a DRM-free per-article PDF through the library's acquisition, the DOI https://doi.org/10.1007/BF01898138 resolving to the article's page; 289,950 bytes. No notice is printed on the scanned pages; the publisher's article page (DOI 10.1007/BF01898138, read 2026-10-02) shows "© Akadémiai Kiadó" under Rights and permissions behind a paywall and names no Open Access or Creative Commons license, every other right reserved.

Read status: claims checked for the introduction (the Erdős--Taylor result, Erdős's question as the paper states it, and the announcement of the result), Theorem 1 with its conditions (1) and (2), Corollary 1 and Corollary 2 (p. 237), the note on refining the sequence and the opening of the proof, with its choice (3) of $n_0$ and of $\varepsilon$ (p. 238), the closing conclusion and the "Added in proof" (p. 241), each read clause by clause on the page images of PDF pp. 1, 2 and 5 on 2026-09-22; the reference list and the received date (p. 241) were read on the same page image. The proof of the first part of Theorem 1 (pp. 238--239) was read in full on the page images and its nested-interval construction was followed but not checked; Lemmas 1 and 2 and the proof of the Hausdorff-dimension part (pp. 239--241) were read on the page images for structure only. Nothing here is independently reviewed.

Contents

• Introduction (p. 237, page image). For a sequence $(q_n)_{n\in\mathbb N^*}$ of positive reals with $q_{n+1}/q_n\ge\lambda$ for some $\lambda>1$ and all $n$, Erdős and Taylor [1] proved that the set of $x$ in any interval $[a,b]$ ($a<b$) such that $(q_nx)$ is not equidistributed mod 1 has Hausdorff dimension 1, a set the paper notes has measure zero. Erdős's question as the paper poses it (p. 237, quoted): "P. Erdős has asked if there exists a real number $x\in[a,b]$ such that the sequence $(q_nx)_{n\in\mathbb N^*}$ is not everywhere dense mod 1." The paper observes that the answer is trivially yes when $\lambda>2$, and announces that it is yes for every $\lambda>1$, that the exceptional $x\in[a,b]$ form a set of Hausdorff dimension 1, and that both follow from a more precise result (Theorem 1), which covers, for instance, the sequence $(x^n)_{n\in\mathbb N^*}$. The paper states Erdős's question without an irrationality clause and with the multiplier confined to an arbitrary interval. Thomas [2] is cited for the earlier result that the $x\in[a,b]$ with $(vx^n)$ not equidistributed mod 1 form a set of dimension 1, for every real $v$.
• Theorem 1 (p. 237, page image; quoted in full on its page). For an interval $[a,b]$ and continuous $\varphi_n\colon[a,b]\to\mathbb R$, monotonic and differentiable on $(a,b)$ with non-vanishing derivatives, such that for some reals $1<\lambda\le\mu$ and every $\xi\in(a,b)$, $n\in\mathbb N^*$, $\lambda\le|\varphi'_{n+1}(\xi)|/|\varphi'_n(\xi)|\le\mu$ (1): there is $x\in[a,b]$ with $(\varphi_k(x))$ not everywhere dense mod 1. If moreover some $\tau\ge0$ has $(\operatorname{Log}|\varphi'_n(\xi)|-\operatorname {Log}|\varphi'_n(\xi')|)\le\tau|\varphi_n(\xi)-\varphi_n(\xi')|$ for all $\xi,\xi'\in(a,b)$ and $n$ (2), the set of such $x$ has Hausdorff dimension 1.
• Corollary 1 (p. 237, quoted): "Let $(q_n)_{n\in\mathbb N^*}$ be a sequence of real positive numbers such that there exists $\lambda>1$ with $q_{n+1}/q_n\ge \lambda$ for all $n$, and let $[a,b]$ be an interval in $\mathbb R$. Then the set of real numbers $x\in[a,b]$ such that the sequence $(q_nx)$ is not everywhere dense mod 1, has Hausdorff dimension 1." Corollary 2 (p. 237, quoted): "Let $v$ be a real number. The set of real numbers $x$ belonging to any interval $[a,b]$ such that the sequence $(vx^n)_{n\in\mathbb N^*}$ is not everywhere dense mod 1, has Hausdorff dimension 1." The note opening p. 238 explains why Corollary 1 needs only a lower bound on the ratios: "we can if necessary refine the sequence $(q_n)$ so that $\lambda\le q_{n+1}/q_n\le\lambda^2$ for all $n$", so that (1) holds with $\mu=\lambda^2$ for $\varphi_n(x)=q_nx$; the refined sequence contains the original, so a multiplier that works for it works for the original.
• Proof of Theorem 1, first part (pp. 238--239, page images). The proof establishes a quantitative form of the first assertion (p. 238): some $\varepsilon>0$ and some $x\in[a,b]$ satisfy $\|\varphi_n(x)\|\ge\varepsilon$ for all sufficiently large $n\in\mathbb N^*$, where $\|z\|=\min_{k\in\mathbb Z}|z-k|$ is the distance from $z\in\mathbb R$ to the nearest integer; the same $\varepsilon$ serves to avoid any prescribed sequence of closed intervals of radius $\varepsilon$ mod 1 for all sufficiently large $n$, by replacing $\varphi_n$ with $\varphi_n+\alpha_n$. The construction: $n_0$ a positive integer with $\lambda^{n_0}\ge2n_0+1$ (3) and $\varepsilon=\mu^{1-2n_0}/2$; after removing at most finitely many terms so that $|\varphi_{n_0}(b)- \varphi_{n_0}(a)|\ge2$, and taking the $\varphi_n$ increasing, the claim becomes that some $x\in[a,b]$ has $\|\varphi_n(x)\|\ge\varepsilon$ for all $n$. With $F_n=\{x\in[a,b]:\|\varphi_n(x)\|\ge\varepsilon\}$ and $G_N=\bigcap _{1\le n\le N}F_n$, integers $K_s$ are chosen inductively so that $\varphi^{-1}_{(s+1)n_0}([K_s,K_s+1])\subset G_{sn_0}$ and the preimages are nested; the counting step (p. 239) shows that at most $2(n_0-1)$ of the unit intervals $[K,K+1]$ inside $\varphi_{(s+1)n_0}(I_{s-1})$ fail to lie in $\varphi_{(s+1)n_0}(G_{sn_0})$, while that image has length at least $\lambda^{n_0}(1-2\varepsilon)\ge\lambda^{n_0}-1$, so by (3) an admissible $[K_s,K_s+1]$ exists. A filing observation, not a review verdict: the paper prints no bound for $\varepsilon$ in terms of $\lambda-1$; for Corollary 1 with $\mu=\lambda^2$ and $\lambda=1+\delta$, the least $n_0$ satisfying (3) is of order $\delta^{-1}\log(1/\delta)$ and the printed $\varepsilon=\mu^{1-2n_0}/2=\lambda^2/(2\lambda^{4n_0})$ is then of order $\delta^4/\log^4(1/\delta)$ (an authored reading of the displayed choices), where Peres and Schlag attribute a separation $c\epsilon^4|\log\epsilon|^{-1}$ jointly to de Mathan and Pollington, three logarithmic factors stronger; the discrepancy is recorded and not resolved here.
• Proof of Theorem 1, Hausdorff-dimension part (pp. 239--241, page images, structure only). Lemma 1 (p. 239): for nested finite families $\mathcal J_s$ of disjoint closed intervals with $\mathcal J_0=\{[a,b]\}$ (4), each interval of $\mathcal J_{s-1}$ containing at least two of $\mathcal J_s$ (5), a separation $d(J,J')\ge\delta|I|$ between distinct children of $I$ (6), and $|I|^\alpha\le\sum_{J\subset I}|J|^\alpha$ (7), the intersection $C$ has Hausdorff dimension at least $\alpha$; proved through Lemma 2, $\sum|\Omega_i|^\alpha\ge\delta^\alpha(b-a)^\alpha$ for every family of open intervals covering $C$ (8), by induction on the level covered; the paper calls it "a result similar to that of [2] (p. 165, IV, th. I$'''$)" with "a very simple proof". The proof of Theorem 1 then resumes with $\lambda^{n_0}\ge 2n_0+2$, builds the families from the intervals $\varphi^{-1}_{(s+1)n_0} ([K+\varepsilon,K+1-\varepsilon])$, verifies (6) from (1) and (2) through the two-sided estimate (9) with $c=e^\tau$, and verifies (7) from the measure estimate (10), choosing $n_0$ large enough for each $\alpha\in(0,1)$. Conclusion (p. 241, quoted): "the set of real numbers $x\in[a,b]$ such that the sequence $(\varphi_n(x))_{n\in\mathbb N^*}$ is not everywhere dense mod 1 (and, indeed, the set of $x\in[a,b]$ such that the sequence $(\varphi_n(x))_{n\in\mathbb N^*}$ does not have zero as a point of accumulation mod 1) has Hausdorff dimension 1."
• Added in proof and references (p. 241, page image). Quoted: "Added in proof (April 8, 1980). Similar results concerning sequences $(q_nx)$ were obtained independently by A. D. Pollington, Illinois J. Math., 23 (1979), 511--515", the paper filed as pollington_1979_density_sequence_n_k_xi, whose own p. 511 credits de Mathan in the same terms. The two references are listed above.

Compiled scope

The paper is compiled at statement depth for the results Problem 464 consumes: Theorem 1 and Corollary 1 (p. 237), read on the page images and quoted on their result pages, with the proof's explicit separation $\|\varphi_n(x)\|\ge\varepsilon$ (p. 238) and the closing accumulation-point form of the conclusion (p. 241). Corollary 2 is recorded as a statement read on the page image, with its own page. The nested-interval construction was followed and not checked, and the dimension argument was read for structure only. Nothing here is independently reviewed.

Bears on. #464: Corollary 1 (printed p. 237, PDF p. 1; quoted above and on its page), that for positive reals $q_n$ with $q_{n+1}/q_n\ge\lambda>1$ for all $n$ and any interval $[a,b]$ the $x\in[a,b]$ for which $(q_nx)$ is not everywhere dense mod 1 form a set of Hausdorff dimension 1, is the other original solution of the corrected formulation of that page, independent of Pollington's: with $q_n=n_k$ and $\lambda=1+\epsilon$ it gives multipliers $\theta$ with $(\theta n_k)$ not dense modulo $1$, and the proof (p. 238) gives $\|\theta n_k\|\ge\varepsilon>0$ for all but finitely many $k$. The paper poses Erdős's question without the irrational clause (p. 237); a set of Hausdorff dimension 1 is uncountable and so contains irrationals, and for an irrational $\theta$ and integers $n_k$ the finitely many excepted terms have $\|\theta n_k\|>0$, so $\inf_k\|\theta n_k\|>0$ (two authored lines the problem page states). The introduction's phrase "not everywhere dense mod 1" is the one the site's thread quotes in correcting the problem's wording. The paper prints no separation bound in terms of $\lambda-1$; the filing observation above records what its displayed choices give.

Results.

• Theorem 1 (p. 237): for $\varphi_n$ continuous on $[a,b]$ and monotonic and differentiable on $(a,b)$, with $\lambda\le|\varphi'_{n+1}|/|\varphi'_n|\le\mu$, $1<\lambda\le\mu$, some $x\in[a,b]$ has $(\varphi_k(x))$ not everywhere dense mod 1; under the Lipschitz condition (2) on $\operatorname{Log}|\varphi'_n|$, the set of such $x$ has Hausdorff dimension 1.
• Corollary 1 (p. 237): for $q_{n+1}/q_n\ge\lambda>1$ and any interval $[a,b]$, the $x\in[a,b]$ with $(q_nx)$ not everywhere dense mod 1 form a set of Hausdorff dimension 1; the answer to Erdős's question.
• Corollary 2 (p. 237): for every real $v$ and any interval $[a,b]$, the $x\in[a,b]$ with $(vx^n)$ not everywhere dense mod 1 form a set of Hausdorff dimension 1; no problem page consumes it.

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