Davenport 1963 theorem uniform distribution

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conjecture_p3: Davenport and Erdős's conjecture that the multiples of almost every alpha > 0 are uniformly distributed relative to any increasing sequence of positive reals z_j tending to infinity with z_(j+1)/z_j -> 1, dropping the monotone-gap hypothesis of the known case; the paper proves it only when O(N^(2 - delta)) of the z_j lie below N.

theorem: Davenport and Erdős's theorem that the multiples of almost every alpha > 0 hit a union of non-overlapping intervals of positive density as often as its measure predicts, provided O(N^(2 - delta)) intervals start at or below N; with the corollary that the multiples of almost every alpha > 0 are uniformly distributed relative to a sequence z_j with z_(j+1)/z_j -> 1 of that sparseness, no monotonicity of the gaps required.

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H. Davenport and P. Erdős, A theorem on uniform distribution, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963), 3--11 (MR 29 #4750; Zbl 122,59). The site's key DaEr63 for Problem 492.

The copy read for this card is the Rényi archive's OmniPage scan, nine pages (printed pp. 3--11 are PDF pp. 1--9; printed p. $n$ is PDF p. $n-2$), with a noisy text layer; the statements below were read on the rendered page images. No notice is printed on the scan's first or last pages; the hosting archive's site footer "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only." (https://users.renyi.hu/~p_erdos/, read 2026-10-02) speaks for the site, not the paper; the journal has no publisher page or DOI for this edition, so the publisher's page was not consulted and no Crossref license is recorded; the term is unstated.

Read status: claims checked for the definitions and the account of the earlier results (printed p. 3), the Theorem, the deduction (9), footnote 4 and Khintchine's question (10) (p. 4), and the general conjecture (p. 5), read clause by clause on the page images; the proof (Sections 2--6, pp. 5--10) was read for its structure and not checked; nothing here is independently reviewed.

LeVeque introduced a more general concept of uniform distribution than distribution modulo 1: for a sequence $z_1<z_2<\cdots$ of positive reals with $z_n\to\infty$ and each $0<\lambda<1$, let $F(N)$ count the positive integers $k\le N$ with $k\alpha$ in one of the intervals (2) $(z_j,z_j+\lambda(z_{j+1}-z_j))$; if $F(N)/N\to\lambda$ for each $\lambda$, the sequence (1) $\alpha,2\alpha,3\alpha,\ldots$ is uniformly distributed relative to $\{z_j\}$ ($z_j=j$ gives distribution modulo 1). The authors assume (3) $z_{j+1}/z_j\to1$, remarking that without it (1) is uniformly distributed relative to $\{z_j\}$ for no $\alpha$ at all. They state the earlier result as a consequence of LeVeque's work and its supplement by Davenport and LeVeque: "provided $z_{j+1}-z_j$ is monotonic (in the wide sense), the sequence (1) is uniformly distributed relative to $\{z_j\}$ for almost all $\alpha>0$" (p. 3). They conjecture this holds without the monotonicity and prove it when the $z_j$ are not very dense: the Theorem (p. 4), for non-overlapping intervals $(x_j,y_j)$ with $x_j\to\infty$ and $I(Z)$ the measure of their part in $(0,Z)$: if (6) $I(Z)\gg Z$ and (7) the number $X(N)$ of $j$ with $x_j\le N$ satisfies $X(N)\ll N^{2-\delta}$ for some fixed $\delta>0$, then (8) $\alpha F_\alpha(N)/I(N\alpha)\to1$ as $N\to\infty$ for almost all $\alpha>0$, where $F_\alpha(N)$ counts $k\le N$ with $k\alpha$ in one of the intervals. Taking (9) $x_j=z_j$, $y_j=z_j+\lambda(z_{j+1}-z_j)$ gives $I(Z)/Z\to\lambda$, and the authors deduce, with no monotonicity assumption, that "the sequence (1) is uniformly distributed relative to $\{z_j\}$ for almost all $\alpha$, provided that the number of $z_j<N$ is $\ll N^{2-\delta}$" (p. 4). They conjecture the theorem holds without (7), are unsure how far (6) can be relaxed, and cannot disprove that $I(Z)\to\infty$ suffices; a footnote says (6) can be relaxed somewhat if (7) is strengthened. The paper then draws attention to a different unsolved question, Khintchine's (p. 4): for a Lebesgue measurable $S\subseteq(0,1)$ of measure $m(S)$ and $F_\alpha(N,S)$ the number of $k\le N$ whose $k\alpha$ has fractional part in $S$, is (10) $F_\alpha(N,S)/N\to m(S)$ for almost all $\alpha$ in $(0,1)$?, and to the general conjecture (p. 5) $\sum_{k\le N}f(k\alpha)/I(N\alpha)\to\alpha^{-1}$ for bounded measurable nonnegative $f$ with $I(Z)=\int_0^Zf\to\infty$, which contains both; the authors say they can contribute nothing toward proving or disproving either conjecture. Khintchine's question is not Problem 492, which is LeVeque's question above. The Russian summary (p. 11) restates the Theorem with $X(N)/N^{2-\delta}$ bounded.

Source: https://users.renyi.hu/~p_erdos/1963-01.pdf.

Bears on. #492: the Theorem and the deduction (9) (printed p. 4, PDF p. 2, page image) are the site's "Davenport and Erdős [DaEr63] proved it is true if $a_n\gg n^{1/2+\epsilon}$": for real sequences $z_j$ with $z_{j+1}/z_j\to1$ and $\ll N^{2-\delta}$ terms below $N$, equivalently $z_j\gg j^{1/2+\epsilon}$, the multiples of almost every $\alpha>0$ are uniformly distributed relative to $\{z_j\}$. The conjecture (p. 3), with $z_j=a_j$, asserts the affirmative answer to the problem's corrected Statement (real sequences): uniform distribution relative to $\{z_j\}$ for almost all $\alpha>0$ under $z_{j+1}/z_j\to1$ alone; p. 3 attributes the monotone-gap case to LeVeque and Davenport--LeVeque, and the paper proves the conjecture only under the counting condition.

Results.

• Conjecture (p. 3): under $z_{j+1}/z_j\to1$, the sequence $\alpha,2\alpha,\ldots$ is uniformly distributed relative to $\{z_j\}$ for almost all $\alpha>0$ without the requirement that the gaps $z_{j+1}-z_j$ be monotonic.
• Theorem (p. 4): if $I(Z)\gg Z$ and $X(N)\ll N^{2-\delta}$ for a fixed $\delta>0$, then $\alpha F_\alpha(N)/I(N\alpha)\to1$ as $N\to\infty$ for almost all $\alpha>0$.
• Deduction (9) (p. 4): if at most $O(N^{2-\delta})$ of the $z_j$ lie below $N$, then for almost every $\alpha>0$ the multiples $\alpha,2\alpha,\ldots$ are uniformly distributed relative to $\{z_j\}$, with no monotonicity assumption.
• Khintchine's problem (10) (p. 4), restated as open: for measurable $S\subseteq(0,1)$, is $F_\alpha(N,S)/N\to m(S)$ for almost all $\alpha$? A different problem from Problem 492.

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