Erdos 1949 strong law large numbers

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remark_p52: Erdős's statements, given without proof, that some f and lacunary n_k have sums of f(n_k x) exceeding N (log log N)^{1/2-eps} by an unbounded factor almost everywhere, while every such sum is o(N (log N)^{1/2+eps}) almost everywhere.

theorem_1: Erdős's construction of a 1-periodic f with mean zero and unit mean square and a lacunary sequence n_k for which the averages of f(n_k x) have limit superior infinity for almost all x, with the variant (5) whose mean-square Fourier tail is below 1/(log log log n)^eps.

theorem_2: Erdős's sharpening of Kac, Salem and Zygmund: if the mean-square Fourier tail of f is O(1/(log log n)^{2+eps}) for some eps > 0, then the averages of f(n_k x) along every lacunary sequence tend to 0 for almost all x.

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P. Erdős: On the strong law of large numbers, Trans. Amer. Math. Soc. 67 (1949), 51--56 MR 11,375c; Zentralblatt 34,72.

For a 1-periodic $f$ with $\int_0^1f=0$ and $\int_0^1f^2=1$, and a sequence with $n_{k+1}/n_k>c>1$, Theorem 1 exhibits an $f$ and an $n_k$ such that for almost all $x$ the averages $\frac1N\sum_{k\le N}f(n_kx)$ have limit superior $\infty$ (p. 51, display (3)), answering no to the question whether the strong law of large numbers proved by Kac, Salem and Zygmund under a Fourier tail condition holds for every such $f$. Theorem 2 sharpens Kac, Salem and Zygmund: the mean-square tail condition $\int_0^1(f-\phi_n(f))^2=O(1/(\log\log n)^{2+\epsilon})$ for some $\epsilon>0$ already forces the averages to tend to $0$ almost everywhere (p. 51, display (4)). Erdős also states, without proof, that a slight modification of the construction of Theorem 1 gives an $f$ and $n_k$ for which the divergence persists although $\int_0^1(f-\phi_n(f))^2<1/(\log\log\log n)^{\epsilon}$, with $\epsilon$ unspecified (pp. 51--52, display (5)), and notes a gap between (4) and (5). He further states, again without proof, that some $f$ and $n_k$ have $\limsup_N N^{-1}(\log\log N)^{-1/2+\epsilon}\sum_{k\le N}f(n_kx)=\infty$ almost everywhere (display (6)), whereas normalizing by $N(\log N)^{1/2+\epsilon}$ always gives limit $0$ (display (7)), and remarks that the constructed $f$ is unbounded, so the bounded case stays open (p. 52). The construction of Theorem 1 sums Rademacher functions $r_m(x)$ over blocks with weights $(A_k(v_k-u_k))^{-1/2}$ and takes for $n_k$ the integers $2^m$ whose exponents $m$ lie in carefully spaced intervals (pp. 52--55); the proof of Theorem 2 is a sketch by correlation bounds and a dyadic covering (pp. 55--56).

Source: https://users.renyi.hu/~p_erdos/1949-09.pdf. No notice is printed on pp. 51--52 or 55--56; the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, read 2026-10-02); the publisher's issue page could not be read on 2026-10-02 (https://www.ams.org/journals/tran/1949-067-01/ redirected to a page holding only site navigation), and the publisher's copyright policy page (https://www.ams.org/publications/authors/ctp, read 2026-10-02) states that the "AMS permits the noncommercial use of its copyrighted works for educational purposes only, such as to quote brief passages or to copy small portions of content for personal use in teaching or research" and names Creative Commons licenses only for its open-access series, every other right reserved.

Read status: claims checked for the setting, Theorems 1 and 2 and displays (3) to (7), read clause by clause on the page images of the print; the proof of Theorem 1 and the sketch of Theorem 2 followed for structure. Displays (5), (6) and (7) are stated in the paper without proof. A second reader checked the result pages' statements, hypotheses, labels and pages against the print; the proofs were not independently reviewed. Result pages: theorem_1, theorem_2 and remark_p52.

Bears on. #995: displays (6) and (7) (p. 52) state without proof, for $f$ normalized as in the paper, a lower bound $N(\log\log N)^{1/2-\epsilon}$ for the growth of $\sum_{k\le N}f(n_kx)$ attained by some $f$ and $n_k$ and an upper bound $o(N(\log N)^{1/2+\epsilon})$ for all; they do not answer the problem's $o(N\sqrt{\log\log N})$ question. #996: Theorem 2 (p. 51) proves the strong law under a tail of order $(\log\log n)^{-2-\epsilon}$ in mean square, stronger than the problem's $\log\log\log n$ condition, and the variant (5) (pp. 51--52) states without proof a divergent example with tail below $(\log\log\log n)^{-\epsilon}$ in mean square, $\epsilon$ unspecified; the paper notes the gap between them and decides nothing about the problem.

Results.

• Theorem 1 (p. 51) and the variant (5) (pp. 51--52): some normalized $f$ and lacunary $n_k$ have averages of $f(n_kx)$ with limit superior $\infty$ almost everywhere.
• Theorem 2 (p. 51): the tail condition (4) gives the strong law (2) along every lacunary sequence.
• Remarks (6) and (7) (p. 52): growth bounds for the sums, stated without proof.

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