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Boon Suan Ho, Counterexamples for lacunary dilates via dyadic spike blocks, arXiv:2604.18535 (v1 of 20 April 2026, v2 of 21 April 2026), answers the example question of Problem 995 in the negative. Theorem 1.6 of v2 (Theorem 1.5 of v1, for $p=2$ only) constructs, for every $2\le p<\infty$, a real mean-zero $f\in L^p(\mathbb T)$ and a lacunary integer sequence $(n_j)$ with $n_{j+1}/n_j\ge2$ such that, for almost every $x$,

$$\limsup_{N\to\infty}\frac{\sum_{j\le N}f(n_jx)}{N(\log N)^{1/p-\varepsilon}} =+\infty\qquad\text{for every }\varepsilon>0.$$

At $p=2$ the denominator $N(\log N)^{1/2-\varepsilon}$ grows faster than $N\sqrt{\log\log N}$, so the partial sums of this $f$ are not $o(N\sqrt{\log\log N})$ almost everywhere, and Section 7 of v2 (Section 6 of v1) draws that conclusion for the problem. Remark 7.1 of v2 adds the elementary upper bound $\sum_{j\le N}f(n_jx)=O(N(\log N)^{1/p+\varepsilon})$ almost everywhere for every increasing sequence and every $f\in L^p$, so the $L^p$ growth scale is fixed up to the $\varepsilon$ gap; at $p=2$ this upper bound is the one Erdős had proved. The construction places rare positive spikes on thin dyadic cylinders so that the lacunary averages see long positive runs, while a deterministic floor prevents cancellation from the other stages. The paper's acknowledgements say that GPT-5.4 Pro was used to explore proof strategies, test intermediate formulations and assist with exposition, and that the author verified the arguments. The statements are those of v2 on its [[../library/analysis/ho_2026_counterexamples_lacunary_dilates_via_dyadic_spike/_index|library card]]; no independent review of the proofs is recorded.

Covers. The example question: for a lacunary sequence and a mean-zero $f\in L^2$ the sums $\sum_{k\le N}f(\{\alpha n_k\})$ need not be $o(N\sqrt{\log\log N})$ for almost all $\alpha$, since Theorem 1.6 of v2 at $p=2$ gives a pair for which the sums exceed $N(\log N)^{1/2-\varepsilon}$ infinitely often almost everywhere. The problem's first sentence asks to estimate the growth without naming a target; the author wrote in the thread that, with Erdős's upper bound, the result places the worst-case growth at $N(\log N)^{1/2+o(1)}$, and a thread participant replied that this may amount to a full solution. Whether that determination answers "estimate the growth" is a matter of reading, so this page claims only the example question, and the growth reading is recorded on the problem page.

Depends on. Nothing in this wiki; the result rests on the preprint alone.

Standing. The author announced the result in the problem's forum thread on 21 April 2026, naming GPT-5.4 Pro's assistance, posted a revised version (v2) to arXiv the same day and announced it in the thread on 22 April 2026; no claim had been filed on the site's proof-claims tab as of 2026-10-07, the site's label is OPEN and its commentary does not name the paper, the arXiv record lists no journal reference (as of 2026-10-07), and no reviewer is named, so the claim stays claimed. The same paper's negative answer to Problem 996 is recorded on its page for that problem.