Almost everywhere divergence of lacunary subsequences of partial sums of Fourier series (Q643839)

For a complex-valued integrable function on \(\mathbb{T}=\mathbb{R} / (2 \pi \mathbb{Z})\) the modulus of continuity is defined by \[ \omega(f,\delta)_1 = \sup_{0 \leq h \leq \delta} \int_{\mathbb{T}} |f(x+h) - f(x)| ~dx. \] Generally, a function \(\omega\) is called a modulus of continuity if it is a continuous nondecreasing, semi-additive function for which \(\omega(0)=0\). Let \(H_1^\omega\) denote the set of functions for which for all \(\delta\) we have \(\omega(f,\delta)_1 \leq \omega(\delta)\). \textit{N. Y. Antonov} [``Almost everywhere divergent subsequences of Fourier sums of functions from \(\varphi (L) \cap H_1^\omega\)'', Math. Notes 85, No. 4, 484--495 (2009); translation from Mat. Zametki 85, No. 4, 502--515 (2009; Zbl 1217.42008)] proved that for any lacunary sequence \((n_k)_{k \geq 1}\) for which \[ \frac{1}{\log k} = \mathcal{O} \left( \omega(1/n_k) \right) \quad \text{as \(k \to \infty\)} \] there exists a function \(f \in H_1^\omega\) for which the partial sums \(S_{n_k}(f,x)\) are a.e. divergent along the subsequence \((n_k)_{k \geq 1}\). On the other hand, for any arbitrary sequence \((n_k)_{k \geq 1}\), if \[ \sum_{k=1}^\infty \frac{\omega(1/n_k)}{k} < \infty, \tag{1} \] then for all functions \(f \in H_1^\omega\) the subsequence of partial sums converges a.e. In the present paper, the author proves that under some weak regularity conditions on \(\omega\) condition (1) is also necessary in the case of lacunary sequences. More precisely, if for a lacunary sequence \((n_k)_{k \geq 1}\) \[ \sum_{k=1}^\infty \frac{\omega(1/n_k)}{k} = \infty, \] then for some function \(f \in H_1^\omega\) the partial sums \(S_{n_k}(f,x)\) are a.e. divergent along the subsequence \((n_k)_{k \geq 1}\). It should be emphasized that this result is almost optimal (only requiring some mild regularity conditions). The proof is based on Antonov's method, but also uses some new ideas.