Counterexamples for lacunary dilates via dyadic spike blocks

Abstract: We construct lacunary counterexamples for two problems of Erdős on the pointwise behavior of dilates $f(n_jx)$ on the circle. The method is a stagewise dyadic spike-block construction: each stage adds a small $L<sup>2$ block and inserts many independent lacunary trials arranged so that a hit on one central spike creates a large partial average, while separation of binary scales keeps the old, active, and future blocks under control. First, we show that very weak Fourier-tail assumptions do not ensure pointwise boundedness of normalized lacunary averages. If $ω$ is decreasing, $Nω(N)<sup>2$ is eventually increasing, and $A^{{1/2}ω(\exp(\exp(κA\log} A)))\to\infty$ for every $κ&gt;0$, then there are a mean-zero $f\in L^2(\mathbb T)$ and a lacunary sequence $(n_j)$ such that $|f-S_Nf|<em>2\llω(N)$ but $$\limsup</em>{N\to\infty} N^{{-1}\sum_{j\le} N} f(n_jx)=+\infty$$ for almost every $x$. In particular, $ω(N)=(\log\log\log N)<sup>{-c}$ gives a negative answer to a question of Erdős, and $ω(N)=(\log\log N)<sup>{-c}$, $0&lt;c\&lt;1/2$, shows Matsuyama's positive range is sharp up to the endpoint. Second, we construct a mean-zero $L^2$ function and a lacunary sequence for which, for every $$\varepsilon\&gt;0$$, $$\limsup_{N\to\infty}\sum_{j\le N}f(n_jx)/(N(\log N)<sup>{1/2-\varepsilon})=+\infty$$ almost everywhere, disproving Erdős's proposed $o(N\sqrt{\log\log N})$ bound. We also include a bounded small-set companion construction recovering lacunary sweeping-out behavior in this dyadic dilation model.