Almost everywhere divergent subsequences of Fourier sums of functions from φ(L) ∩ H ω 1

For a gap sequence of natural numbers {n k } =1 ∞ , for a nondecreasing function φ: [0,+∞) → [0,+∞) such that φ(u) = o(u ln ln u) as u → ∞, and a modulus of continuity satisfying the condition (ln k)−1 = O(ω(n −1 )), we present an example of a function F ∈ φ(L) ∩ H 1 with an almost everywhere divergent subsequence {S n k (F, x)} of the sequence of partial sums of the trigonometric Fourier series of the function F.