Erdos 1975 problems results diophantine approximations ii

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question_p96_generalized_progression: Erdős's 1975 definition of a generalized arithmetic progression as the sequence of integer parts of t alpha + beta with real alpha > 1 and beta, and his question whether the complement of every sufficiently fast-growing sequence contains an infinite one.

question_p96_lacunary_density: Erdős's 1975 question whether every integer sequence with ratios n_{k+1}/n_k > c > 1 admits an irrational alpha for which the fractional parts of n_k alpha are not everywhere dense, with the Erdős--Taylor dimension-one result he reports beside it; the printed origin of Problem 464.

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P. Erdős, 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 chapter continues Erdős's Problems and results on diophantine approximations, Compositio Math. 16 (1964), 52--65, which it cites on its first page as (I). Cited as [Er75i] on the problem pages; the site's reference text names the volume without the chapter. The citing sources filed here identify the chapter in the volume: Dubickas's reference [14] gives pp. 89--99, Pollington's reference [2] and Peres and Schlag's reference [5] give the volume, and the references of Erdős's 1982 paper Some of my favourite problems which recently have been solved (p. 63) give pp. 89--97, a range the scan shows is short by two pages.

The copy read for this card is an 11-page OmniPage 12 scan of the chapter alone (created 18 November 2004, per the file's metadata), with an OCR text layer that locates passages and garbles the displays. Printed pp. 89--99 = PDF pp. 1--11 (printed p. $n$ is PDF p. $n-88$): PDF pp. 2--11 carry the printed numbers 90--99 at the top, and PDF p. 1, the chapter's first page, carries the title and author and no number. The scan prints neither the volume's title nor its year nor its series; those come from the citing sources above. Provenance: a scan obtained in a survey download of September 2026; the download URL was not recorded; 1,029,329 bytes. No notice is printed on the chapter scan; the publisher's chapter page (DOI 10.1007/BFb0074258, read 2026-10-02) shows "© 1975 Springer-Verlag" as subscription content and names no Open Access or Creative Commons license, every other right reserved.

Read status: claims checked for the lacunary density question and the generalized arithmetic progression question of p. 96, read clause by clause on the page image of PDF p. 8 on 2026-09-22. The first page (p. 89), the closing problems (pp. 97--98) and the references (p. 99) were read on the page images of PDF pp. 1, 9, 10 and 11; §§ 2--6 (pp. 89--96) were first read in the text layer only, for their section structure and the names in them, and their summaries below were checked against the page images of PDF pp. 1--8 on 2026-10-07. The chapter states questions and reports results of others; it proves nothing, so there is no proof to check, and nothing here is independently reviewed.

Contents

• Opening (p. 89, page image). Erdős says that the chapter first reports the progress made on the problems of his earlier paper, which he refers to as (I), and then states a few new questions at the end; he adds that he has not worked on the subject recently and asks forgiveness for any omitted reference. Six numbered sections follow, each closing with its own references, and the new questions form an unnumbered tail.
• § 1 (p. 89, page image). For a sequence $x_1,x_2,\ldots$ in $(0,1)$ and the exponential sums $\sum_{j\le n}e^{2\pi ikx_j}$, Erdős's conjecture that the limiting moduli $A_k$ satisfy $\limsup_k A_k=\infty$, which he proved, and Clunie's stronger result that $|A_k|>c\sqrt k$ for infinitely many $k$; whether $|A_k|>ck$ holds for infinitely many $k$ is asked, with Clunie's example of a sequence with $|A_k|\le k$ for all $k$.
• § 2 (pp. 89--91, page images). The discrepancy $D(x_1,\ldots,x_n)$ of van der Corput; Schmidt's $\max_{m\le n}D(x_1,\ldots,x_m)>10^{-2}\log n$, sharpening van Aardenne-Ehrenfest and Roth; Erdős's question (1) whether some sequence has $\limsup_n|N_n(a,b)-(b-a)n|<\infty$ for every interval (the display prints $\limsup_nN_n(a,b)<\infty$, without the term $(b-a)n$), answered in the negative by Schmidt, with Schmidt's bounds (2) and (3) and the gap between them; the Hecke--Ostrowski boundedness for $x_n=\{n\beta\}$, $\beta$ irrational, when the interval length is $\{m\beta\}$, and the converse conjectured by Erdős and Szüsz and proved by Kesten; the discrepancy questions of (I) in higher dimensions resolved by Schmidt, and Philipp's results on $\{n_k\alpha\}$ for $n_{k+1}/n_k>c>1$. References to van Aardenne-Ehrenfest 1949, Roth 1955, Schmidt's "On irregularities of distribution I to IX", Kesten 1968, Kesten and Sós 1968 and Philipp.
• § 3 (pp. 91--92, page images). The notation $(k\alpha)=k\alpha-[k\alpha]$ for the fractional part is set here. Khintchine's conjecture that $f_n(E,\alpha)/n\to m(E)$ for almost all $\alpha$ and every measurable $E$, disproved by Marstrand (1970) with a closed set $E$ for which the upper and lower limits are $1$ and $0$ for almost all $\alpha$; well-distributed sequences after Hlawka and Petersen; Erdős's claim in (I) that $(p_n\alpha)$, $p_n$ the $n$th prime, is not well distributed for some irrational $\alpha$, whose proof he "was not able to reconstruct" (p. 92), and his belief that it fails for every $\alpha$.
• § 4 (pp. 92--94, page images). Erdős's necessary and sufficient condition for almost all $\alpha$ to have infinitely many solutions of $|\alpha-a/n_k|<\varepsilon/n_k^2$ with $(a,n_k)=1$ along a sequence $n_1<n_2<\cdots$; the "old conjecture" (2), $\sum_{q=1}^\infty\varepsilon_q\varphi(q)/q=\infty$, which contains that theorem and which Erdős hopes his method will prove (the scan states neither the theorem's condition nor the conjecture in full: p. 92 ends at "is that" and p. 93 opens with display (2)); Cassels's property of sequences, Haight's construction (in his thesis) of a sequence for which the averaged ratios tend to $0$, and Erdős's results from (I) on $\varphi(n_1,\ldots,n_{k-1};n_k)/n_k$. References to Erdős, J. Number Theory 2 (1970), and Philipp's Memoir 114.
• § 5 (pp. 94--95, page images). LeVeque's uniform distribution modulo a sequence $A$ with $a_{n+1}/a_n\to1$: whether $n\alpha$ is uniformly distributed modulo $A$ for almost all $\alpha$, proved by Davenport and Erdős for $a_n>n^{1/2+\epsilon}$ and disproved in general by Schmidt (1969); Schmidt's question of a set of infinite measure in which no quotient of two elements is an integer, answered by Haight and by Szemerédi, with Erdős's question how fast its measure up to $x$ can grow; a question of Haight on the integer multiples of a set of positive measure, and Haight's bound $m(S)<9$ for a set $S$ of positive numbers containing at most one multiple of each $x>1$, which Erdős asks whether to improve to $m(S)\le3/2$, best possible if true. References to Schmidt 1969, Erdős and Davenport 1963, Haight 1970, Szemerédi 1971 and Lekkerkerker 1958.
• § 6 (pp. 95--96, page images). Heilbronn's problem of estimating $f_3(n)$, with $f_k(n)$ the largest minimum area of a $k$-gon among $n$ points in the unit circle: Roth's and Schmidt's upper bounds for $f_3(n)$, ending with Roth's $f_3(n)<c/n^{1+\alpha}$, $\alpha=(9-65^{1/2})/8$, and the question whether $f_3(n)<c/n^2$; Schmidt's $f_4(n)>c/n^{3/2}$ and conjecture $f_k(n)=o(1/n)$; an observation with Roth that for $n$ points in the unit sphere the smallest-angle method gives $f_3^{(3)}(n)<c/n^{1/2}$ and the smallest-diameter method $c/n^{2/3}$, and the question whether $f_3^{(3)}(n)=o(1/n^{2/3})$. References to Roth 1972 and Schmidt 1971--1972.
• The disconnected problems (pp. 96--98, page images). The tail opens on p. 96 by announcing a few disconnected problems and poses the first in these words: "Let $n_1<n_2<\cdots$ be an infinite sequence of integers satisfying $n_{k+1}/n_k>c>1$. Is it true that there always is an irrational $\alpha$ for which the sequence $(n_k\alpha)$ is not everywhere dense?" Erdős adds that he and Taylor proved that the set of $\alpha$ for which $(n_k\alpha)$ is not uniformly distributed is a set of Hausdorff dimension $1$. The parentheses are the chapter's fractional part notation of § 3; the Erdős--Taylor result is the paper listed as reference [1] on p. 99, and the text prints no citation marker. The next paragraph, also p. 96, calls the sequence $[t\alpha+\beta]$, $t$ an integer, a generalized arithmetic progression for real $\alpha>1$ and $\beta$, and asks: "Let $\{n_k\}$ tend to infinity sufficiently fast. Is it true that the complement of $\{n_k\}$ contains an infinite generalized arithmetic progression?" Then (pp. 96--97) the density $f(x)$ of the integers with $\varphi(n)/n<x$, shown to exist by Schoenberg and purely singular by Erdős (the p. 99 references [4] and [2]), and whether its derivative can exist and differ from $0$, or take any given value; (p. 97) for points $z_n$ on the unit circle, whether $A_n=\max_{|z|=1}\prod_{i\le n}|z-z_i|$ is unbounded, with Hayman's example showing $A_n\le n$ is possible, and the general version for a closed set $D$ without interior points comparing $A_n(D)$ with the Chebyshev quantity $E_n(D)$ over monic polynomials with roots in $D$; (pp. 97--98) for a primitive sequence $1<a_1<\cdots$ of integers, Erdős's $\sum_k1/(a_k\log a_k)<c$ (1) and Behrend's bound (2), sharpened with Sárközy and Szemerédi (the p. 99 reference [3]), and the question whether for real $1<a_1<\cdots$ with $|a_j-\ell a_i|\ge1$ for all integers $i,j,\ell$ (3), the bounds (1) and (2) still follow; Erdős could not prove that (3) gives $\liminf A(x)/x=0$, Haight proved $\lim A(x)/x=0$ under rational independence, and Besicovitch's result shows (3) does not force it for integers.
• References (p. 99, page image). Four items: [1] Erdős and Taylor, Proc. London Math. Soc. 7 (1957), 598--615; [2] Erdős, Amer. J. Math. 61 (1939), 722--725; [3] Erdős, Sárközy and Szemerédi, Number Theory Colloquium, János Bolyai Math. Soc., North Holland (1968), 36--49; [4] Schoenberg, Math. Zeitschrift 28 (1928), 171--190. The author's address is the Mathematical Institute of the Hungarian Academy of Sciences.

Compiled scope

The chapter is compiled at statement depth for the two questions of p. 96 that Problem 464 consumes, each on its own result page (listed under Results below) and quoted above from the page image. The rest of the chapter is mapped from its text layer and checked against the page images of pp. 89--99; no other question is compiled, and nothing is independently reviewed.

Bears on. #464: p. 96 poses the question in Erdős's own words (the lacunary density question): for a sequence with $n_{k+1}/n_k>c>1$, whether some irrational $\alpha$ makes $(n_k\alpha)$ "not everywhere dense", with $(x)$ the fractional part (§ 3, p. 91). It is the printed origin that the site's key Er75i names, that Pollington's introduction quotes with the ratio condition written $\ge\alpha>1$, and that Erdős's 1982 paper restates as "not everywhere dense". The printed wording carries the irrationality clause and asks about density of the fractional parts modulo $1$, not of the distances to the nearest integer, as the problem page's corrected Statement does; the strict ratio bound $>c>1$ and the site's $n_{k+1}\ge(1+\epsilon)n_k$ describe the same class of sequences. The same page reports the Erdős--Taylor theorem that the multipliers $\alpha$ for which $(n_k\alpha)$ is not uniformly distributed form a set of Hausdorff dimension one, which does not answer the question. The next paragraph poses the generalized arithmetic progression question, not a statement of Problem 464, beside which the problem page records the results of Graham and Sós and of Pollington from Erdős and Graham's 1980 book (its printed p. 18). The chapter proves nothing about either question.

Results. The lacunary density question (p. 96); the generalized arithmetic progression question (p. 96).

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