Counterexamples for lacunary dilates via dyadic spike blocks

TL;DR

The paper constructs dyadic lacunary counterexamples to Erdős's problems on pointwise behavior of dilates, showing limitations of positive theorems and providing new functions with divergent averages.

Contribution

It introduces dyadic spike block constructions that produce counterexamples for Erdős's pointwise dilate problems, especially at critical exponents.

Findings

Counterexamples show Matsuyama's theorem cannot extend to c=1/2.

Constructs functions in L^p with divergent lacunary averages for p≥2.

Answers Erdős Problems #995 and #996 negatively.

Abstract

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.