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It is a classical result that dyadic partial sums of the Fourier series offunctions $f \in L^p(\mathbb{T})$ converge almost everywhere for $p \in (1,\infty)$. In 1968, E. A. Bredihina established an analogous result forfunctions belonging to the Stepanov space of almost periodic functions $S^2$whose Fourier exponents satisfy a natural separation condition. Here, themaximal operator corresponding to dyadic partial summation of almost periodicFourier series is bounded on the Stepanov spaces $S^{2^k}$, $k \in \mathbb{N}$for functions satisfying the same condition; Bredihina's result follows as aconsequence. In the process of establishing these bounds, some general resultsare obtained which will facilitate further work on operator bounds andconvergence issues in Stepanov spaces. These include a boundedness theorem forthe Hilbert transform and a theorem of Littlewood--Paley type. An improvementof "$S^{2^k}$, $k \in \mathbb{N}$" to "$S^p$, $p \in (1, \infty)$" is also seento follow from a natural conjecture on the boundedness of the Hilberttransform.

Pointwise Convergence of Dyadic Partial Sums of Almost Periodic Fourier Series