= Decay of coefficients of an almost-everywhere convergent trigonometric series
If symmetric partial sums $\sum_{r=-n}^n a_re^{irt}$ converge almost everywhere on the circle, then $a_n,a_{-n}\to0$. Their successive differences $T_n=a_ne^{int}+a_{-n}e^{-int}$ tend to zero almost everywhere. If $M_n=|a_n|+|a_{-n}|$ failed to tend to zero, choose a subsequence with $M_n\ge\varepsilon$. The normalized $T_n/M_n$ are bounded by one and tend to zero almost everywhere, so <dominated convergence> makes their squared integrals tend to zero. <Fourier orthogonality> instead makes those integrals $(|a_n|^2+|a_{-n}|^2)/M_n^2\ge1/2$, a contradiction.
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