  This article establishes a real-variable argument for Zygmund's theorem on
almost everywhere convergence of strong arithmetic means of partial sums of
Fourier series on $\mathbb{T}$, up to passing to a subsequence. Our approach
extends to, among other cases, functions that are defined on $\mathbb{T}^d$,
which allows us to establish an analogue of Zygmund's theorem in higher
dimensions.
