Counterexamples for lacunary dilates via dyadic spike blocks

We construct dyadic lacunary counterexamples for two problems of Erdős 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 $f\in {\bigcap }_{1\le q<\mathrm{\infty }}{L}^q(\mathbb{T})$ and a sequence $n_j=2^{m_j}$, $n_{j+1}\mathrm{/}{n}_j\ge 2$, such that $$\mathrm{\parallel }f-S_{N}f{\mathrm{\parallel }}_2\ll (\log \log N{)}^{-1\mathrm{/}2},\underset{N\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{1}{N}\sum _{j\le N}f(n_{j}x)=+\mathrm{\infty }$$ for almost every $x$. Thus Matsuyama’s positive theorem at exponent $c>1\mathrm{/}2$ cannot be extended to the endpoint $c=1\mathrm{/}2$, and Erdős Problem #996 has a negative answer. A second choice of parameters gives, for every $2\le p<\mathrm{\infty }$, functions $f\in L^p(\mathbb{T})$ with $$\underset{N\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\sum _{j\le N}f(n_{j}x)}{N(\log N{)}^{1\mathrm{/}p-\epsilon }}=+\mathrm{\infty }(\epsilon >0)$$ almost everywhere; the case $p=2$ answers Erdős Problem #995. We also include a bounded small-set companion construction.