Decay of coefficients of an almost-everywhere convergent trigonometric series

If symmetric partial sums ${\sum }_{r=-n}^n{a}_r{e}^{irt}$ converge almost everywhere on the circle, then $a_n,a_{-n}\to 0$. Their successive differences $T_n=a_n{e}^{int}+a_{-n}{e}^{-int}$ tend to zero almost everywhere. If $M_n=|a_n|+|a_{-n}|$ failed to tend to zero, choose a subsequence with $M_n\ge \epsilon$. 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 $(|a_n{|}^2+|a_{-n}{|}^2)/M_n^2\ge 1/2$, a contradiction.