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Claim. D. A. Raikov (printed D. Raikoff), On some arithmetical properties of summable functions, Mat. Sb. (N.S.) 1(43) (1936), no. 3, 377--384, in Russian with an English summary. Theorem 1 of the paper states that if $f$ is an arbitrary integrable function of period $1$ and $a>1$ is a natural number, then $\frac1n\sum_{k=0}^{n-1}f(a^kx)\to\int_0^1f(t)\,dt$ for almost every $x$ as $n\to\infty$. Theorem 2 gives convergence in mean, not almost everywhere, for an arbitrary increasing sequence of natural numbers. Applied to the lacunary sequence $n_k=a^k$, Theorem 1 gives the conclusion of Problem 996 for every $f\in L^2([0,1])\subset L^1([0,1])$, with no condition on its Fourier tail, so every exponent $C$ works for these sequences. The journal record gives the year and no day, so this page carries the first of January.

Covers. The question for $n_k=a^k$ with an integer $a\ge2$: the averages converge to the integral for almost every $\alpha$ for every $f\in L^2([0,1])$, so the Fourier-tail condition is unnecessary and every $C$ works; nothing about other lacunary sequences.

Depends on. No page of this wiki.

Acceptance. Refereed: Matematicheskii Sbornik 1(43) (1936). The site's commentary credits Raikov with this case, but the site labels the problem OPEN, so no curator acceptance is listed. The theorem is stated from the paper's statement of Theorem 1; its proof has not been reproduced here.