Almost everywhere divergence of lacunary subsequences of partial sums of Fourier series

If an increasing sequence $\{n_m\}$ of positive integers and a modulus of continuity $\omega$ satisfy the condition $\sum_{m=1}^\infty\omega(1/n_m)/m<\infty$, then it is known that the subsequence of partial sums $S_{n_m}(f,x)$ converges almost everywhere to $f(x)$ for any function $f\in H_1^\omega$. We show that this sufficient convergence condition is close to a necessary condition for a lacunary sequence $\{n_m\}$.