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A continuous complex-valued function can have uniformly bounded Fourier partial sums which fail to converge at one point. Normalize harmonic sine polynomials by their harmonic coefficient sums, modulate them into disjoint high-frequency intervals and choose their degrees so that the complete norms are summable. At zero, complete blocks vanish while their midpoint prefixes have a common nonzero value. The frequency-separated Fourier block series gives a uniform bound on every partial sum as well as two distinct subsequential values.

Every null subset of the circle lies in the divergence set of the Fourier series of some continuous complex-valued function. One can arrange unbounded Fourier partial sums there. The compact-set Fourier amplification lemma, compact batching of a small open set and frequency-separated Fourier block series construct the function by a summable series of small blocks. The result covers nonclosed and dense null sets; it does not require the divergence set to equal the originally specified set.