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Statement

Setting (p. 51). $f$ satisfies $f(x+1)=f(x)$, $\int_0^1f(x)\,dx=0$ and $\int_0^1f(x)^2\,dx=1$; $n_1<n_2<\cdots$ is any sequence with $n_{k+1}/n_k>c>1$; $\phi_n(f)$ is the $n$th partial sum of the Fourier series of $f$. Kac, Salem and Zygmund proved the conclusion below under their condition, the paper's display (1), $\int_0^1(f(x)-\phi_n(f))^2=O(1/(\log n)^{\epsilon})$ for some $\epsilon>0$.

Theorem 2 (p. 51). Assume that for some $\epsilon>0$

$$\int_0^1(f(x)-\phi_n(f))^2=O\Big(\frac{1}{(\log\log n)^{2+\epsilon}}\Big).$$

This is the paper's display (4). Then the paper's display (2) holds: for almost all $x$

$$\lim_{N\to\infty}\frac1N\sum_{k=1}^Nf(n_kx)=0.$$

The paper adds (p. 52) that it seems probable that (4) can be replaced by $1/(\log\log n)^{\eta}$, but that much sharper methods would be needed. It proves nothing in that direction.

Proof pointer

Pp. 55--56, a sketch the paper labels as such. For $j-i=r$ one has $n_j/n_i>c^r$, so by (4) and the Cauchy--Schwarz inequality $\int_0^1f(n_ix)f(n_jx)\,dx<c_1/(\log r)^{1+\epsilon/2}$. Summing gives $\int_0^1\big(\sum_{k=s}^{s+N}f(n_kx)\big)^2=O(N^2/(\log N)^{1+\epsilon/2})$, so by Chebyshev's inequality the set where such a block sum exceeds $AN$ has measure at most $c/A^2(\log N)^{1+\epsilon/2}$. A dyadic covering of the partial sums by blocks of decreasing length, a method the paper attributes to Hobson, Plancherel, Rademacher and Menchoff, makes the total measure of the exceptional sets summable, and outside them $\lvert\sum_{k\le m}f(n_kx)\rvert<2\delta m$.

Read depth

Claims checked: the setting, (1), (4), Theorem 2 and the remark on replacing (4) were read clause by clause on the page images of the print, and the sketch on pp. 55--56 was followed. A second reader checked the statement, hypotheses, label and page against the print; the sketch was not independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Kac, Salem and Zygmund, Trans. Amer. Math. Soc. 63 (1948), 235--243, and the Hobson--Plancherel--Rademacher--Menchoff method (Rademacher, Math. Ann. 87 (1922), 117--121).

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 996: Theorem 2 proves the problem's conclusion, for $f$ normalized as in the paper, under the tail condition (4), a power of $\log\log n$ with exponent above $2$ in mean square. The problem asks about a power of $\log\log\log n$, so Theorem 2 does not decide it.