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We provide a near-complete classification of the Lorentz spaces$\Lambda_{\varphi}$ for which the sequence $\{S_{n}\}_{n\in \mathbb{N}}$ ofpartial Fourier sums is almost everywhere convergent along lacunarysubsequences. Moreover, under mild assumptions on the fundamental function$\varphi$, we identify $\Lambda_{\varphi}:= L\log\log L\log\log\log\log L$ asthe \emph{largest} Lorentz space on which the lacunary Carleson operator isbounded as a map to $L^{1,\infty}$. In particular, we disprove a conjecturestated by Konyagin in his 2006 ICM address. Our proof relies on a newlyintroduced concept of a "Cantor Multi-tower Embedding," a special geometricconfiguration of tiles that can arise within the time-frequency tiledecomposition of the Carleson operator. This geometric structure plays animportant role in the behavior of Fourier series near $L^1$, being responsiblefor the unboundedness of the weak-$L^1$ norm of a "grand maximal countingfunction" associated with the mass levels.

The pointwise convergence of Fourier Series (I). On a conjecture of Konyagin