---

Boon Suan Ho, Counterexamples for lacunary dilates via dyadic spike blocks, arXiv:2604.18535, gives the negative answer. Version 1 (20 April 2026) already does so in its Corollary 1.3, with an $f\in L^2$ depending on $C$, and reaches only exponents $c<1/2$ of $\log\log N$; the statements and numbering below are those of version 2 (21 April 2026), which strengthens the main result to the endpoint. Theorem 1.1 constructs a real mean-zero $f$ in every $L^p(\mathbb T)$ with $p<\infty$ and a lacunary integer sequence with $n_{j+1}/n_j\ge 2$ such that $\|f-S_Nf\|_2\ll(\log\log N)^{-1/2}$ while $\limsup_N N^{-1}\sum_{j\le N}f(n_jx)=+\infty$ for almost every $x$. Corollary 1.4 draws the consequence for the problem: since $(\log\log N)^{-1/2}=o((\log\log\log N)^{-C})$ for every $C>0$, the same pair satisfies the triple-logarithmic Fourier-tail condition at every exponent and its averages diverge almost everywhere, so no absolute $C$ exists. Corollary 1.5 notes that the exponent range $c>1/2$ in Matsuyama's positive theorem is sharp. Theorem 1.7 is a bounded companion: a set $E$ of measure below any prescribed $\varepsilon$ and a lacunary sequence along which the averages of $\mathbf 1_E$ have limit superior $1$ almost everywhere, with no Fourier-tail estimate asserted for it. The construction places rare positive spikes on thin dyadic cylinders so that 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 and assist with exposition and that the author verified the arguments. The paper's [[../library/analysis/ho_2026_counterexamples_lacunary_dilates_via_dyadic_spike/_index|library card]] records the statements; the proofs were not checked here.

Standing. The result was not filed on the site's proof-claims tab; a thread comment of 27 April 2026 points to the preprint. The site's label is unchanged (OPEN) and its commentary does not name the author, the arXiv record lists no journal reference, and no reviewer is named, so the claim stays claimed. A later, independent construction with a bounded observable, which credits this paper with the first disproof, is recorded on Yang's page.