AbstractIn this paper, we investigate the relationship between pointwise convergence of the arithmetic means corresponding to the subsequence of partial Fourier sums $$(S_{k_j}f: j\in \mathbb {N})$$ ( S k j f : j ∈ N ) (for $$f\in L^1(\mathbb {T})$$ f ∈ L 1 ( T ) ) and the structure of the chosen subsequence of the sequence of natural numbers $$(k_j: j\in \mathbb {N})$$ ( k j : j ∈ N ) . More precisely, the problem we deal with is to provide necessary and sufficient conditions for a subsequence $$\mathcal {N}$$ N of $$\mathbb {N}$$ N that has the following property: for any subsequence $$\mathcal {N^{\prime }} = (k_j: j\in \mathbb {N})$$ N ′ = ( k j : j ∈ N ) of $$\mathcal {N}$$ N and any $$f\in L^1(\mathbb {T})$$ f ∈ L 1 ( T ) one has $$\frac{1}{N}\sum _{j=1}^N S_{k_j}f(x) \rightarrow f(x)$$ 1 N ∑ j = 1 N S k j f ( x ) → f ( x ) for a.e. $$x\in \mathbb {T}$$ x ∈ T . A direct corollary of this paper’s main theorem is that there exists a subsequence $$(k_j)$$ ( k j ) of the sequence of natural numbers and an integrable function f such that the arithmetic means of $$S_{k_j}f$$ S k j f do not converge to f almost everywhere. This is a negative answer to a question that originated in an article by Zalcwasser in 1936 Zalcwasser (Stud. Math. 6, 82–88 (1936)) for some increasing sequences $$(k_j)$$ ( k j ) of natural numbers.