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 ω satisfy the condition Σ =1 ∞ ω(1/n m )/m < ∞, then it is known that the subsequence of partial sums $S_{n_m } \left( {f,x} \right)$ converges almost everywhere to f(x) for any function f ∈ H 1 . We show that this sufficient convergence condition is close to a necessary condition for a lacunary sequence {n m }.