We provide a near-complete classification of the Lorentz spaces $\Lambda_{ \varphi}$ for which the sequence $\{ S_{ n}\}_{ n\in \mathbb{ N}}$ of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function $\varphi$, we identify $\Lambda_{ \varphi}:= L\log\log L\log\log\log\log L$ as the \emph{ largest} Lorentz space on which the lacunary Carleson operator is bounded as a map to $L^{ 1,\infty}$. In particular, we disprove a conjecture stated by Konyagin in his 2006 ICM address. Our proof relies on a newly introduced concept of a "Cantor Multi-tower Embedding," a special geometric configuration of tiles that can arise within the time-frequency tile decomposition of the Carleson operator. This geometric structure plays an important role in the behavior of Fourier series near $L^1$, being responsible for the unboundedness of the weak-$L^1$ norm of a "grand maximal counting function" associated with the mass levels.
Categories: math.CA
