Almost everywhere convergence of Fourier series in L log L

Theorem 1.1: for every complex-valued $f\in L\log L(\mathbb T)$ there is a null set $N_f$ with $\lim_{N\to\infty}S_Nf(x)=f(x)$ for $x\notin N_f$, along the full sequence. The key estimate is a good-set bound $\int_{\{M\rho\le e^{4h}\}}\sup_N|S_N\rho|\le Cd$ for bounded densities of height $e^d$, summed over height layers. The paper also deduces a weak-$L^1$ maximal inequality in Luxemburg norm. It does not address spaces strictly between $L\log L$ and $L^1$ (where Konyagin's divergence results apply below $L\sqrt{\log L}$), nor lacunary or higher-dimensional analogues.

Theorem 1.1: for every complex-valued $f\in L\log L(\mathbb T)$ there is a null set $N_f$ with $\lim_{N\to\infty}S_Nf(x)=f(x)$ for $x\notin N_f$, along the full sequence. The key estimate is a good-set bound $\int_{\{M\rho\le e^{4h}\}}\sup_N|S_N\rho|\le Cd$ for bounded densities of height $e^d$, summed over height layers. The paper also deduces a weak-$L^1$ maximal inequality in Luxemburg norm. It does not address spaces strictly between $L\log L$ and $L^1$ (where Konyagin's divergence results apply below $L\sqrt{\log L}$), nor lacunary or higher-dimensional analogues.