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Statement

Setting (p. 51). Throughout the paper $f$ is a function on $-\infty<x<\infty$ satisfying $f(x+1)=f(x)$, $\int_0^1 f(x)\,dx=0$, $\int_0^1 f(x)^2\,dx=1$; $n_1<n_2<\cdots$ is a sequence with $n_{k+1}/n_k>c>1$; and $\phi_n(f)$ is the $n$th partial sum of the Fourier series of $f$. The paper's display (2), the conclusion of Kac, Salem and Zygmund and the strong law for $f(n_kx)$, is $\lim_{N\to\infty}\frac1N\sum_{k=1}^N f(n_kx)=0$ for almost all $x$.

Theorem 1 (p. 51). There exist an $f$ satisfying these conditions and a sequence $n_k$ with $n_{k+1}/n_k>c>1$ such that for almost all $x$

$$\limsup_{N\to\infty}\frac{1}{N}\Big(\sum_{k=1}^{N}f(n_kx)\Big)=\infty.$$

This is the paper's display (3). It answers no to the question, recorded in the paper, whether (2) holds for every such $f$; a footnote (p. 51) attributes the case $n_k=2^k$, where (2) does hold, to Raikov.

Variant (5) (pp. 51--52). The paper states, without proof, that a slight modification of the construction gives an $f$ and a sequence $n_k$ for which (3) holds and

$$\int_0^1(f(x)-\phi_n(f))^2<\frac{1}{(\log\log\log n)^{\epsilon}}.$$

The print does not quantify $\epsilon$ in (5). The paper notes a gap between (5) and the hypothesis (4) of Theorem 2.

The function of Theorem 1 is unbounded, and the paper records (p. 52) that whether (2) holds for every bounded $f$ remains open.

Proof pointer

Pp. 52--55, proof of Theorem 1. With $r_m$ the Rademacher functions and $u_k,v_k,A_k$ growing fast, $f$ is a sum over blocks $u_k<m\le v_k$ of $r_m(x)/(A_k(v_k-u_k))^{1/2}$, with $\sum_k1/A_k=1$, so that $f$ satisfies the conditions of the setting. The sequence $n_k$ consists of the integers $2^m$ with $m$ in $j_k$ disjoint intervals $I_t^{(k)}$ of lengths $l_t^{(k)}$ for each $k$, where $j_k$ grows with $A_k$. On each interval the $k$th block contributes a Rademacher sum; a large deviation estimate (cited from Erdős, Ann. of Math. 43 (1942)) and the independence of the intervals make one of the $j_k$ averages exceed $3c$ with probability close to $1$, and Chebyshev's inequality controls the earlier and later blocks. This gives limit superior above every $c$ almost everywhere, hence (3).

Read depth

Claims checked: the setting, Theorem 1, (5) and the boundedness remark were read clause by clause on the page images of the print, and the proof on pp. 52--55 was followed for structure. The modification giving (5) is not written out in the paper. A second reader checked the statement, hypotheses, label and page against the print; the proof was not independently reviewed.

Dependencies

None in the corpus. External input: the large deviation lower bound for sums of Rademacher functions, cited from Erdős, Ann. of Math. 43 (1942), p. 420, formula (0.7), with a correction printed in the paper's footnote 4.

Source. P. Erdős, On the strong law of large numbers, Trans. Amer. Math. Soc. 67 (1949), 51--56; the edition read is named on the source card.

Bears on

• Problem 995: Theorem 1 shows that the sums $\sum_{k\le N}f(n_kx)$ need not be $o(N)$ almost everywhere for a lacunary sequence and a mean-zero $f$ with $\int_0^1f^2=1$. It does not address the problem's example question about $o(N\sqrt{\log\log N})$; the sharper growth statements are the remarks (6) and (7).
• Problem 996: (5) states, without proof, that some $f$ and $n_k$ satisfy (3) and have mean-square Fourier tail below $(\log\log\log n)^{-\epsilon}$, with $\epsilon$ unquantified; the paper notes a gap between (4) and (5) and does not claim an answer to the problem's question.