Counterexamples for lacunary dilates via dyadic spike blocks

We construct dyadic lacunary counterexamples for two problems of Erd\H{o}s on pointwise behavior of dilates on the circle. The main device is a dyadic spike block: rare positive spikes create long positive runs in the lacunary averages, while a deterministic lower floor prevents cancellation from the remaining stages. The endpoint construction gives a mean-zero $f\in\bigcap_{1\le q<\infty}L^q(\mathbb T)$ and a sequence $n_j=2^{m_j}$, $n_{j+1}/n_j\ge2$, such that $$\|f-S_Nf\|_2\ll (\log\log N)^{-1/2}, \qquad \limsup_{N\to\infty} \frac1N\sum_{j\le N}f(n_jx)=+\infty$$ for almost every $x$. Thus Matsuyama's positive theorem at exponent $c>1/2$ cannot be extended to the endpoint $c=1/2$, and Erd\H{o}s Problem #996 has a negative answer. A second choice of parameters gives, for every $2\le p<\infty$, functions $f\in L^p(\mathbb T)$ with $$\limsup_{N\to\infty} \frac{\sum_{j\le N}f(n_jx)} {N(\log N)^{1/p-\varepsilon}} =+\infty \qquad(\varepsilon>0)$$ almost everywhere; the case $p=2$ answers Erd\H{o}s Problem #995. We also include a bounded small-set companion construction.